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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

sweety439 2022-02-05 05:57

numbers n contain no digit 0 in all bases 3<=b<=n-1: [URL="https://oeis.org/A069575"]A069575[/URL]
numbers n contain no digit z in all bases 3<=b<=n: [URL="https://oeis.org/A337536"]A337536[/URL]

(both sequences are conjectured to be finite and full, the largest such numbers are 619 and 256, respectively)

(since in base 2 all numbers (except 0) contain the digit z (= 1 in base 2), and all numbers which are not Mersenne numbers [URL="https://oeis.org/A000225"]A000225[/URL] contain the digit 0, and any number n contain the digit 0 in base n (since the representation is 10), and any number n contain the digit z (= n in base n+1) in base n+1 (since the representation is single-digit number "z" = n), one cannot be better than this)

sweety439 2022-02-05 09:43

family 1{0}z = {1z, 10z, 100z, 1000z, 10000z, 100000z, 1000000z, 10000000z, 100000000z, 1000000000z, ...}

family z{0}1 = {z1, z01, z001, z0001, z00001, z000001, z0000001, z00000001, z000000001, z0000000001, ...}

family y{z} = {yz, yzz, yzzz, yzzzz, yzzzzz, yzzzzzz, yzzzzzzz, yzzzzzzzz, yzzzzzzzzz, yzzzzzzzzzz, ...} (since the numbers must be > base, the number "y" is not counted)

family {z}1 = {z1, zz1, zzz1, zzzz1, zzzzz1, zzzzzz1, zzzzzzz1, zzzzzzzz1, zzzzzzzzz1, zzzzzzzzzz1, ...} (since the numbers must be > base, the number "1" is not counted)

family 1{0}2 = {12, 102, 1002, 10002, 100002, 1000002, 10000002, 100000002, 1000000002, 10000000002, ...}

family 2{0}1 = {21, 201, 2001, 20001, 200001, 2000001, 20000001, 200000001, 2000000001, 20000000001, ...}

family 1{z} = {1z, 1zz, 1zzz, 1zzzz, 1zzzzz, 1zzzzzz, 1zzzzzzz, 1zzzzzzzz, 1zzzzzzzzz, 1zzzzzzzzzz, ...} (since the numbers must be > base, the number "1" is not counted)

family {z}y = {zy, zzy, zzzy, zzzzy, zzzzzy, zzzzzzy, zzzzzzzy, zzzzzzzzy, zzzzzzzzzy, zzzzzzzzzzy, ...} (since the numbers must be > base, the number "y" is not counted)

family 1{0}1 = {11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, ...}

family {1} = {11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, ...} (since the numbers must be > base, the number "1" is not counted)

family {z0}z1 = {z1, z0z1, z0z0z1, z0z0z0z1, z0z0z0z0z1, z0z0z0z0z0z1, z0z0z0z0z0z0z1, z0z0z0z0z0z0z0z1, z0z0z0z0z0z0z0z0z1, z0z0z0z0z0z0z0z0z0z1, ...}

sweety439 2022-02-07 15:21

[QUOTE=sweety439;594923]These factor pattern can show that such families contain no primes > base:

Reference: the [URL="https://en.wikipedia.org/wiki/Divisibility_rule"]divisibility rule[/URL] for base b:

* For prime p dividing b, the number is divisible by p if and only if the last digit of this number is divisible by p.
* For prime p dividing b-1, the number is divisible by p if and only if the sum of the digits of this number is divisible by p.
* For prime p dividing b+1, the number is divisible by p if and only if the [URL="https://en.wikipedia.org/wiki/Alternating_sum"]alternating sum[/URL] of the digits of this number is divisible by p. (this can also show that all [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]palindromic primes[/URL] in any base b have an odd number of digits, the only possible exception is "11" in base b)

(in these examples, only list the numbers > base)

Example 1: base 10, family 4{6}9 (formula: (14*10^(n+1)+7)/3) ([URL="http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
49 = 7 * 7
469 = 7 * 67
4669 = 7 * 667
46669 = 7 * 6667
466669 = 7 * 66667
4666669 = 7 * 666667
[/CODE]

Example 2: base 10, family 28{0}7 (formula: 28*10^(n+1)+7) ([URL="http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
287 = 7 * 41
2807 = 7 * 401
28007 = 7 * 4001
280007 = 7 * 40001
2800007 = 7 * 400001
28000007 = 7 * 4000001
[/CODE]

Example 3: base 9, family {1} (formula: (9^n-1)/8) ([URL="http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 2 * 5
111 = 7 * 14
1111 = 22 * 45
11111 = 67 * 144
111111 = 222 * 445
1111111 = 667 * 1444
11111111 = 2222 * 4445
111111111 = 6667 * 14444
1111111111 = 22222 * 44445
11111111111 = 66667 * 144444
111111111111 = 222222 * 444445
1111111111111 = 666667 * 1444444
[/CODE]

Example 4: base 9, family 3{8} (formula: 4*9^n-1) ([URL="http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
38 = 5 * 7
388 = 18 * 21
3888 = 58 * 61
38888 = 188 * 201
388888 = 588 * 601
3888888 = 1888 * 2001
38888888 = 5888 * 6001
388888888 = 18888 * 20001
3888888888 = 58888 * 60001
38888888888 = 188888 * 200001
388888888888 = 588888 * 600001
[/CODE]

Example 5: base 8, family 1{0}1 (formula: 8^(n+1)+1) ([URL="http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 3 * 3
101 = 5 * 15
1001 = 11 * 71
10001 = 21 * 361
100001 = 41 * 1741
1000001 = 101 * 7701
10000001 = 201 * 37601
100000001 = 401 * 177401
1000000001 = 1001 * 777001
10000000001 = 2001 * 3776001
100000000001 = 4001 * 17774001
1000000000001 = 10001 * 77770001
[/CODE]

Example 6: base 11, family 2{5} (formula: (5*11^n-1)/2) ([URL="http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
25 = 3 * 9
255 = 2 * 128
2555 = 3 * 919
25555 = 2 * 12828
255555 = 3 * 91919
2555555 = 2 * 1282828
25555555 = 3 * 9191919
255555555 = 2 * 128282828
2555555555 = 3 * 919191919
25555555555 = 2 * 12828282828
[/CODE]

Example 7: base 12, family {B}9B (formula: 12^(n+2)-25) ([URL="http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]formula[/URL])

[CODE]
9B = 7 * 15
B9B = 11 * AB
BB9B = B7 * 105
BBB9B = 11 * B0AB
BBBB9B = BB7 * 1005
BBBBB9B = 11 * B0B0AB
BBBBBB9B = BBB7 * 10005
BBBBBBB9B = 11 * B0B0B0AB
BBBBBBBB9B = BBBB7 * 100005
BBBBBBBBB9B = 11 * B0B0B0B0AB
[/CODE]

Example 8: base 14, family B{0}1 (formula: 11*14^(n+1)+1) ([URL="http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
B1 = 5 * 23
B01 = 3 * 395
B001 = 5 * 22B3
B0001 = 3 * 39495
B00001 = 5 * 22B2B3
B000001 = 3 * 3949495
B0000001 = 5 * 22B2B2B3
B00000001 = 3 * 394949495
B000000001 = 5 * 22B2B2B2B3
B0000000001 = 3 * 39494949495
[/CODE]

Example 9: base 13, family 3{0}95 (formula: 3*13^(n+2)+122) ([URL="http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
395 = 14 * 2B
3095 = 7 * 58A
30095 = 5 * 7A71
300095 = 7 * 5758A
3000095 = 14 * 23A92B
30000095 = 7 * 575758A
300000095 = 5 * 7A527A71
3000000095 = 7 * 57575758A
30000000095 = 14 * 23A923A92B
300000000095 = 7 * 5757575758A
3000000000095 = 5 * 7A527A527A71
[/CODE]

Example 10: base 16, family {4}D (formula: (4*16^(n+1)+131)/15) ([URL="http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
4D = 7 * B
44D = 3 * 16F
444D = D * 541
4444D = 7 * 9C0B
44444D = 3 * 16C16F
444444D = D * 540541
4444444D = 7 * 9C09C0B
44444444D = 3 * 16C16C16F
444444444D = D * 540540541
4444444444D = 7 * 9C09C09C0B
44444444444D = 3 * 16C16C16C16F
444444444444D = D * 540540540541
[/CODE]

Example 11: base 17, family 1{9} (formula: (25*17^n-9)/16) ([URL="http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
19 = 2 * D
199 = B * 27
1999 = 2 * D4D
19999 = AB * 287
199999 = 2 * D4D4D
1999999 = AAB * 2887
19999999 = 2 * D4D4D4D
199999999 = AAAB * 28887
1999999999 = 2 * D4D4D4D4D
19999999999 = AAAAB * 288887
199999999999 = 2 * D4D4D4D4D4D
1999999999999 = AAAAAB * 2888887
[/CODE]

Example 12: base 36, family O{Z} (formula: 25*36^n-1) ([URL="http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
OZ = T * V
OZZ = 4Z * 51
OZZZ = TZ * U1
OZZZZ = 4ZZ * 501
OZZZZZ = TZZ * U01
OZZZZZZ = 4ZZZ * 5001
OZZZZZZZ = TZZZ * U001
OZZZZZZZZ = 4ZZZZ * 50001
OZZZZZZZZZ = TZZZZ * U0001
OZZZZZZZZZZ = 4ZZZZZ * 500001
OZZZZZZZZZZZ = TZZZZZ * U00001
OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001
[/CODE]

Some references of this, see:

[URL="http://www.worldofnumbers.com/wing.htm"]http://www.worldofnumbers.com/wing.htm[/URL] for:

{1}0{1} (base 10): (formula: (10^(2*n+1)-9*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29-9*10%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
101 = 1 * 101 (the only possible prime case)
11011 = 11 * 1001
1110111 = 111 * 10001
111101111 = 1111 * 100001
11111011111 = 11111 * 1000001
1111110111111 = 111111 * 10000001
[/CODE]

{1}2{1} (base 10): (formula: (10^(2*n+1)+9*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29%2B9*10%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
121 = 11 * 11
11211 = 101 * 111
1112111 = 1001 * 1111
111121111 = 10001 * 11111
11111211111 = 100001 * 111111
1111112111111 = 1000001 * 1111111
[/CODE]

{3}2{3} (base 10): (formula: (10^(2*n+1)-3*10^n-1)/3) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29-3*10%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
323 = 17 * 19
33233 = 167 * 199
3332333 = 1667 * 1999
333323333 = 16667 * 19999
33333233333 = 166667 * 199999
3333332333333 = 1666667 * 1999999
[/CODE]

{3}4{3} (base 10): (formula: (10^(2*n+1)+3*10^n-1)/3) ([URL="http://factordb.com/index.php?query=%2810%5E%282*n%2B1%29%2B3*10%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
343 = 7 * 49
33433 = 67 * 499
3334333 = 667 * 4999
333343333 = 6667 * 49999
33333433333 = 66667 * 499999
3333334333333 = 666667 * 4999999
[/CODE]

[URL="http://www.worldofnumbers.com/deplat.htm"]http://www.worldofnumbers.com/deplat.htm[/URL] for:

1{2}1 (base 10): (formula: (11*10^(n+1)-11)/9) ([URL="http://factordb.com/index.php?query=%2811*10%5E%28n%2B1%29-11%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
11 = 11 * 1 (the only possible prime case)
121 = 11 * 11
1221 = 11 * 111
12221 = 11 * 1111
122221 = 11 * 11111
1222221 = 11 * 111111
12222221 = 11 * 1111111
[/CODE]

7{3}7 (base 10): (formula: (22*10^(n+1)+11)/3) ([URL="http://factordb.com/index.php?query=%2822*10%5E%28n%2B1%29%2B11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
77 = 11 * 7
737 = 11 * 67
7337 = 11 * 667
73337 = 11 * 6667
733337 = 11 * 66667
7333337 = 11 * 666667
73333337 = 11 * 6666667
[/CODE]

9{7}9 (base 10): (formula: (88*10^(n+1)+11)/9) ([URL="http://factordb.com/index.php?query=%2888*10%5E%28n%2B1%29%2B11%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
99 = 11 * 9
979 = 11 * 89
9779 = 11 * 889
97779 = 11 * 8889
977779 = 11 * 88889
9777779 = 11 * 888889
97777779 = 11 * 8888889
[/CODE]

9{4}9 (base 10): (formula: (85*10^(n+1)+41)/9) ([URL="http://factordb.com/index.php?query=%2885*10%5E%28n%2B1%29%2B41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
99 = 11 * 9
949 = 13 * 73
9449 = 11 * 859
94449 = 3 * 31483
944449 = 11 * 85859
9444449 = 7 * 1349207
94444449 = 11 * 8585859
944444449 = 13 * 72649573
9444444449 = 11 * 858585859
94444444449 = 3 * 31481481483
944444444449 = 11 * 85858585859
9444444444449 = 7 * 1349206349207
94444444444449 = 11 * 8585858585859
[/CODE]

[URL="http://www.worldofnumbers.com/Appending%201s%20to%20n.txt"]http://www.worldofnumbers.com/Appending%201s%20to%20n.txt[/URL] for:

37{1} (base 10): (formula: (334*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%28334*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
37 = 37 * 1 (the only possible prime case)
371 = 7 * 53
3711 = 3 * 1237
37111 = 37 * 1003
371111 = 13 * 28547
3711111 = 3 * 1237037
37111111 = 37 * 1003003
371111111 = 7 * 53015873
3711111111 = 3 * 1237037037
37111111111 = 37 * 1003003003
371111111111 = 13 * 28547008547
3711111111111 = 3 * 1237037037037
37111111111111 = 37 * 1003003003003
[/CODE]

38{1} (base 10): (formula: (343*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%28343*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
38 = 2 * 19
381 = 3 * 127
3811 = 37 * 103
38111 = 23 * 1657
381111 = 3 * 127037
3811111 = 37 * 103003
38111111 = 233 * 163567
381111111 = 3 * 127037037
3811111111 = 37 * 103003003
38111111111 = 2333 * 16335667
381111111111 = 3 * 127037037037
3811111111111 = 37 * 103003003003
38111111111111 = 23333 * 1633356667
[/CODE]

176{1} (base 10): (formula: (1585*10^n-1)/9) ([URL="http://factordb.com/index.php?query=%281585*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

[CODE]
176 = 11 * 16
1761 = 3 * 587
17611 = 11 * 1601
176111 = 13 * 13547
1761111 = 11 * 160101
17611111 = 7 * 2515873
176111111 = 11 * 16010101
1761111111 = 3 * 587037037
17611111111 = 11 * 1601010101
176111111111 = 13 * 13547008547
1761111111111 = 11 * 160101010101
17611111111111 = 7 * 2515873015873
176111111111111 = 11 * 16010101010101
[/CODE]

Sierpinski number 78557: 78557*2^(n+1)+1, 10011001011011101{0}1 in base 2 (period: 36) ([URL="http://factordb.com/index.php?query=78557*2%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])

Riesel number 509203: 509203*2^n-1, 1111100010100010010{1} in base 2 (period: 24) ([URL="http://factordb.com/index.php?query=509203*2%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factordb[/URL])[/QUOTE]

However, we can find [URL="https://en.wikipedia.org/wiki/Semiprime"]semiprime[/URL] candidate, 3-[URL="https://en.wikipedia.org/wiki/Almost_prime"]almost prime[/URL] candidate, 4-almost prime candidate, etc. for these families, although this is outside of the researching in this thread, e.g.

3{8} in base 9 is 4*9^n-1, we can find n's such that 2*3^n-1 and 2*3^n+1 are both primes
{1} in base 9 is (9^n-1)/8, we can find n's such that (3^n-1)/2 and (3^n+1)/4 or (3^n-1)/4 and (3^n+1)/2 are both primes
1{0}1 in base 8 is 8^(n+1)+1, we can find n's such that 2^n+1 and 4^n-2^n+1 are both primes
8{F} in base 16 is 9*16^n-1, we can find n's such that 3*4^n-1 and 3*4^n+1 are both primes
{F}7 in base 16 is 16^(n+1)-9, we can find n's such that 4^n-3 and 4^n+3 are both primes
1{5} in base 16 is (4*16^n-1)/3, we can find n's such that 2*4^n-1 and (2*4^n+1)/3 are both primes
{C}D in base 16 is (4*16^(n+1)+1)/5, we can find n's such that 2*4^n-2*2^n+1 and (2*4^n+2*2^n+1)/5 or (2*4^n-2*2^n+1)/5 and 2*4^n+2*2^n+1 are both primes
{B}9B in base 12 is 12^(n+2)-25, for even n it has algebraic factors, we can find n's such that 12^n-5 and 12^n+5 are both primes
8{D} in base 14 is 9*14^n-1, for even n it has algebraic factors, we can find n's such that 3*14^n-1 and 3*14^n+1 are both primes
etc.

References: [URL="https://mersenneforum.org/showthread.php?t=22201"]https://mersenneforum.org/showthread.php?t=22201[/URL] [URL="https://mersenneforum.org/showthread.php?t=19209"]https://mersenneforum.org/showthread.php?t=19209[/URL]

Twin prime search (see [URL="http://mersenneforum.org/showthread.php?t=8479"]http://mersenneforum.org/showthread.php?t=8479[/URL] [URL="https://www.primepuzzles.net/problems/prob_049.htm"]https://www.primepuzzles.net/problems/prob_049.htm[/URL] [URL="https://www.rieselprime.de/Related/RieselTwinSG.htm"]https://www.rieselprime.de/Related/RieselTwinSG.htm[/URL] [URL="http://www.noprimeleftbehind.net/gary/twins100K.htm"]http://www.noprimeleftbehind.net/gary/twins100K.htm[/URL] [URL="http://www.noprimeleftbehind.net/gary/twins1M.htm"]http://www.noprimeleftbehind.net/gary/twins1M.htm[/URL]): k*2^n+-1 are twin primes if and only if (k^2)*2^n-1 is semiprime.

[URL="http://www.worldofnumbers.com/Appending%201s%20to%20n.txt"]http://www.worldofnumbers.com/Appending%201s%20to%20n.txt[/URL] also has "semiprimes pattern" of 38{1} = (343*10^n-1)/9, i.e. (7*10^n-1)/3 and (49*10^(2*n)+7*10^n+1)/3 are both primes

New Mersenne Conjecture (see [URL="http://www.hoegge.dk/mersenne/NMC.html"]http://www.hoegge.dk/mersenne/NMC.html[/URL] [URL="https://primes.utm.edu/mersenne/NewMersenneConjecture.html"]https://primes.utm.edu/mersenne/NewMersenneConjecture.html[/URL] [URL="http://www.primenumbers.net/rl/nmc/"]http://www.primenumbers.net/rl/nmc/[/URL]) is consider the n such that 2^n-1 and (2^n+1)/3 are both primes, and they are both primes if and only if (4^n-1)/3 is semiprime.

sweety439 2022-02-08 15:45

For the original (i.e. prime > base is not required) minimal prime in base b=10, 1235607889460606009419 is the smallest prime containing all minimal primes as subsequence (see [URL="https://www.primepuzzles.net/puzzles/puzz_178.htm"]https://www.primepuzzles.net/puzzles/puzz_178.htm[/URL]), and for b=12, 1234456789A04AAA00B0001 (656969693573113867991809 in decimal) is the smallest prime containing all minimal primes as subsequence (see [URL="https://oeis.org/A110600"]https://oeis.org/A110600[/URL]), but for this new minimal prime problem (i.e. start with b+1) ....

In base 10 the minimal set is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, [B]Problem: find the smallest prime containing all these 77 primes as subsequence[/B] (of course, this prime will contain at least 28 0's and at least 11 5's), this problem is equivalently to finding the [URL="https://en.wikipedia.org/wiki/Shortest_common_supersequence_problem"]shortest common supersequence[/URL] of these 77 strings, if this sequence is not prime, then find the next-shortest common supersequence, etc. ([URL="https://algorithm-visualizer.org/dynamic-programming/shortest-common-supersequence"]this[/URL] is an online program to find the shortest common supersequence, but only support 2 strings, and cannot handle this problem (since this problem has 77 strings))

In base 2, the minimal set is {11}, and this prime is clearly 11 (3 in decimal), if primes in the minimal set themselves are not counted, then this prime is 101 (5 in decimal) (this condition (i.e. primes in the minimal set is not counted) will give a different prime only in base b=2, since b=2 is the only base such that there is only one minimal prime (start with b+1)

In base 3, the minimal set is {12, 21, 111}, and this prime is 1121 (43 in decimal), note that 1211 (49 in decimal) is not prime, and (surprisingly) the smallest prime of the form 12{1} is 121111111111111111111111 (24 digits!), which is equal the minimal prime (start with b+1) in base b=9: 544444444444 (Note: although 1112 (41 in decimal) and 2111 (67 in decimal) are primes, but 1112 does not contain 21 as subsequence, and 2111 does not contain 12 as subsequence)

In base 4, the minimal set is {11, 13, 23, 31, 221}, and 12231 is the smallest number containing all these numbers as subsequence (at least one 1 must be after two 2's, also at least one 1 must be after one 3), but it is not prime (it is 429 in decimal), such number must contain >=2 1's, >=2 2's, and >=1 3's, but if all minimal number of digits are realized, then the digit sum is 2*1+2*2+1*3 = 9, thus the number is divisible by 3 (see [URL="https://en.wikipedia.org/wiki/Divisibility_rule"]https://en.wikipedia.org/wiki/Divisibility_rule[/URL], in base b, if d divides b-1, then "n is divisible by d" if and only if "sum of digits of n is divisible by d"), thus, we must add one digits (other than 0 and 3, or the number will be still divisible by 3), we choose 1 and make the number 112231, it equal decimal 1453, a prime, thus 112231 (1453 in decimal) is "this prime" in base 4

sweety439 2022-02-08 16:07

[QUOTE=sweety439;597528]Proving the set of the minimal primes (start with b+1) in base b is S, is equivalent to:

* Prove that all elements in S, when read as base b representation, are primes > b.

* Prove that all proper subsequence of all elements in S, when read as base b representation, which are > b, are composite.

* Prove that all primes > b, when written in base b, contain at least one element in S as subsequence (equivalently, prove that all strings not containing any element in S as subsequence, when read as base b representation, which are > b, are composite).[/QUOTE]

An interesting and difficult problem is find the smallest prime p such that all strings not containing any element in S (S is the set of the minimal primes (start with b+1) in base b) as subsequence, when read as base b representation, which are > b, are divisible by at least one prime <= p (such prime may not exist, as in some bases (such as 8, 9, 12, 14, 16), there are some families which are ruled out as only contain composite numbers by all or partial algebraic factors, in these bases we find the smallest prime p such that all strings not containing any element in S (S is the set of the minimal primes (start with b+1) in base b) as subsequence, when read as base b representation, which are > b, are either "divisible by at least one prime <= p" or "of the special form", the special forms are:

[CODE]
base (b) forms
8 1{0}1
9 {1}, 3{1}, 3{8}, 3{8}35, {8}5
12 {B}9B
14 8{D}, {D}5
16 10{5}, 1{5}, {4}1, 7{3}, 8{5}, 8{F}, B{4}1, {C}D, {C}DD, {F}7
17 1{9}, {9}8
19 1{6}, {6}5, 7{2}, 89{6}
24 3{N}, 5{N}, {6}1, 8{N}
25 {1}, 1{3}, 1{8}, 2{1}, {3}2, 5{1}, 5{8}, 7{1}, {8}3, {8}7, C{1}, F{1}, M{1}, 27{1}
27 1{0}8, 7{Q}, 8{0}1, 9{G}, {D}E, {G}7, {Q}J
32 1{0}1, {1}
33 F{W}, L{4}, {W}H
34 1{B}, 8{X}, G{B}, L{B}, {X}P
36 3{7}, 3{Z}, 8{Z}, 9{5}, G{7}, O{Z}, {Z}B
[/CODE]

The sequence of the smallest prime factor of the numbers in these families is very likely to be unbounded above, thus such p in these bases very unlikely to exist.

Equivalently, finding the smallest prime p such that "the set of the minimal ("the primes > b and <= p" and "the numbers > b not divisible by any prime <= p") in base b" is the same as "the set of the minimal primes (start with b+1) in base b"

In base b=10 such prime is 214741, which is needed to remove the composite number 5(0^24)27, see [URL="http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 5{0}27[/URL] and [URL="http://factordb.com/index.php?query=%285*10%5E%28n%2B1%29-41%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {5}1[/URL]

In base b=2 such prime is 10 (decimal 2), since all numbers > 1 not divisible by 2 (i.e. the odd numbers > 1) have first digit 1 and last digit 1 in base b=2, thus the only minimal prime (start with b+1) 11 (decimal 3) must be a subsequence.

In base b=3 such prime is 10 (decimal 3); first, 10 (decimal 3) is needed, since this prime is needed to remove the numbers 1{0} (i.e. 100, 1000, 10000, 100000, ...); second, for the numbers not divisible by 2 or 3, such number cannot end with 0 (of course also cannot begin with 0) and must have an odd number of 1, thus either 111 (i.e. 3 1's) or 12 or 21 must be a subsequence, unless the number is 1, which is not allowed in this research.

In base b=4 such prime is 3 (decimal 3), which is used to remove the numbers: 21 (decimal 9), all numbers of the form 2{0}1, and the numbers containing only 0 and 3

In base b=5 such prime is 11332432 (decimal 105367), which is used to remove the number 1(0^53)13, see [URL="http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 1{0}13 in base 5[/URL]

In base b=6 such prime is 21 (decimal 13), which is used to remove the number 441 (decimal 169), note that 4041 (decimal 889) is divisible by 7

In base b=7 such prime is 15421 (decimal 4327), see [URL="http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {3}1 in base 7[/URL] and [URL="http://factordb.com/index.php?query=36*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 51{0}1 in base 7[/URL]

In base b=8 such prime does not exist, since the family 1{0}1 has infinite subset whose elements are pairwise coprime but the family 1{0}1 can be ruled out as only contain composite numbers, but if we only consider the numbers not in the family 1{0}1, such prime will be the smallest prime factor of (4*8^217+17)/7 (if it is > 7885303569123738614221) or 7885303569123738614221, which is needed to remove the composites (4^216)7 and (4^116)7, respectively, see [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {4}7 in base 8[/URL]

In base b=9 such prime does not exist, since the families {1}, 3{1}, 3{8}, 3{8}35, {8}5 has infinite subset whose elements are pairwise coprime but the families {1}, 3{1}, 3{8}, 3{8}35, {8}5 can be ruled out as only contain composite numbers, but if we only consider the numbers not in the families {1}, 3{1}, 3{8}, 3{8}35, {8}5, such prime exists, but is very hard to find, since finding this prime requires factoring the large numbers in the families 3{0}11, 2{7}07, 7{6}2

In base b=12 such prime does not exist, since the family {B}9B has infinite subset whose elements are pairwise coprime but the family {B}9B can be ruled out as only contain composite numbers, but if we only consider the numbers not in the family {B}9B, such prime will be 534AB547A0351 (decimal 47113717465069), see [URL="http://factordb.com/index.php?query=4*12%5E%28n%2B2%29%2B91&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 4{0}77 in base 12[/URL] and [URL="http://factordb.com/index.php?query=11*12%5E%28n%2B2%29%2B119&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of B{0}9B in base 12[/URL]

sweety439 2022-02-09 08:15

[QUOTE=sweety439;599662]

[CODE]
base (b) forms
8 1{0}1
9 {1}, 3{1}, 3{8}, 3{8}35, {8}5
12 {B}9B
14 8{D}, {D}5
16 10{5}, 1{5}, {4}1, 7{3}, 8{5}, 8{F}, B{4}1, {C}D, {C}DD, {F}7
17 1{9}, {9}8
19 1{6}, {6}5, 7{2}, 89{6}
24 3{N}, 5{N}, {6}1, 8{N}
25 {1}, 1{3}, 1{8}, 2{1}, {3}2, 5{1}, 5{8}, 7{1}, {8}3, {8}7, C{1}, F{1}, M{1}, 27{1}
27 1{0}8, 7{Q}, 8{0}1, 9{G}, {D}E, {G}7, {Q}J
32 1{0}1, {1}
33 F{W}, L{4}, {W}H
34 1{B}, 8{X}, G{B}, L{B}, {X}P
36 3{7}, 3{Z}, 8{Z}, 9{5}, G{7}, O{Z}, {Z}B
[/CODE]

[/QUOTE]

Note 1: these families are not such families, since although they have full covering set of all or partial algebra factors, but they still have primes for very small lengths:

[CODE]
base (b) forms the only primes in the forms
4 {1} 11
8 {1} 111
16 {1} 11
27 {1} 111
36 {1} 11
[/CODE]

Note 2: these x{y} and {x}y families are not such families, since although they can be ruled out as only contain composite numbers by a full covering set of all or partial algebra factors, but they still have subsequences which are primes (such primes must be repunit primes) since their repeating digit (i.e. y for x{y}, x for {x}y) is 1:

[CODE]
base (b) forms
14 B{1}
16 8{1}
17 5{1}
24 L{1}
33 7{1}, 9{1}
36 O{1}
[/CODE]

Note 3: these families are not such families, since although they can be ruled out as only contain composite numbers by a full covering set of all or partial algebra factors, but they also can be ruled out as only contain composite numbers by covering congruence:

[CODE]
base (b) forms
9 6{1}, 16{1}
14 3{D}
17 3{1}
19 4{9}, 8{3}, G{1}
25 1F{1}
29 2{7}, 2E{7}, 3{S}, 6{1}, 9{4}, C{7}
[/CODE]

sweety439 2022-02-12 20:20

1 Attachment(s)
[QUOTE=sweety439;597067]* Case (2,1):

** [B]21[/B] is prime, and thus the only minimal prime in this family.

* Case (2,2):

** Since 21, 25, 12, 32, 52, [B]272[/B] are primes, we only need to consider the family 2{0,2,4,6,8}2 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}2 are divisible by 2, thus cannot be prime

* Case (2,4):

** Since 21, 25, 14, 34, 74 are primes, we only need to consider the family 2{0,2,4,6,8}4 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}4 are divisible by 2, thus cannot be prime

* Case (2,5):

** [B]25[/B] is prime, and thus the only minimal prime in this family.

* Case (2,7):

** Since 21, 25, 47, 67, 87 are primes, we only need to consider the family 2{0,2,3,7}7 (since any digits 1, 4, 5, 6, 8 between them will produce smaller primes)

*** If there are at least two 3's in {}, then 337 will be a subsequence.

*** If there are exactly one 3's in {}, then there cannot be 7's in {}, otherwise, either 377 or 737 will be a subsequence, thus the family is 2{0,2}3{0,2}7

**** All numbers of the form 2{0,2}3{0,2}7 are divisible by 2, thus cannot be prime

*** If there are no 3's in {}, then the family will be 2{0,2,7}7

**** Since [B]2027[/B] and 272 are primes, we only need to consider the family 2{2}{0,7}7 (since any digits combo 02, 72 between them will produce smaller primes, thus let "d" be the rightmost digit 2 in {}, then all digits before "d" are 2 (cannot be 0 or 7, otherwise 02 or 72 will be in {}, and hence either [B]2027[/B] or 272 will be a subsequence); also, all digits after "d" are 0 or 7, since "d" is the rightmost digit 2, thus the family is 2{2}{0,7}7)

***** Since [B]22227[/B] is prime, we only need to consider the families 2{0,7}7, 22{0,7}7, 222{0,7}7

****** Since [B]2207[/B] is prime, we only need to consider the families 2{0,7}7, 22{7}7, 222{7}7

******* For the 2{0,7}7 family, since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 7

******** If there are only 1 7's, then the form is 2{0}7

********* The smallest prime of the form 2{0}7 is [B]2000000000007[/B]

******** If there are at least 3 7's, then there cannot be any 0 before the 3rd rightmost 7, or [B]20777[/B] will be a subsequence, thus the family is 2{7}7{0}7{0}7

********* Since [B]270707[/B] is prime, we only need to consider the families 2{7}7{0}77 and 2{7}77{0}7

********** All numbers of the form 2{7}7{0}77 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

********** For the 2{7}77{0}7 family, since [B]2770007[/B] is prime, we only need to consider the families 2{7}777, 2{7}7707, 2{7}77007

*********** All numbers of the form 2{7}777 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

*********** The smallest prime of the form 2{7}7707 is [B]27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707[/B] with 687 7's, which can be written as 2(7^686)07 and equal the prime (23*9^688-511)/8 ([URL="http://factordb.com/cert.php?id=1100000002495467486"]primality certificate of this prime[/URL])

*********** All numbers of the form 2{7}77007 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

******* The smallest prime of the form 22{7}7 is [B]22777[/B]

******* All numbers of the form 222{7}7 are divisible by 2 (if the total number of 7's is even) or 5 (if the total number of 7's is odd), thus cannot be prime.

* Case (2,8):

** Since 21, 25, 18, 58, 78, [B]238[/B] are primes, we only need to consider the family 2{0,2,4,6,8}8 (since any digits 1, 3, 5, 7 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6,8}8 are divisible by 2, thus cannot be prime[/QUOTE]

* Case (3,1):

** Since 32, 34, 21, 41, 81, [B]331[/B], [B]371[/B] are primes, we only need to consider the family 3{0,1,5,6}1 (since any digits 2, 3, 4, 7, 8 between them will produce smaller primes)

*** If there are at least two 5's in {}, then 355 will be a subsequence.

*** If there are exactly one 5's in {}, then the form is 3{0,1,6}5{0,1,6}1

**** Since 315, 65, [B]3501[/B], [B]3561[/B] are primes, we only need to consider the family 3{0}5{1}1 (since any digit 1, 6 between (3,5{0,1,6}1) will produce small primes, and any digit 0, 6 between (3{0,1,6}5,1) will produce small primes)

***** Since [B]305111[/B] is prime, we only need to consider the families 35{1}1, 3{0}51, 3{0}511

****** The smallest prime of the form 35{1}1 is [B]351111111[/B]

****** The smallest prime of the form 3{0}51 is [B]30000000000000000000051[/B]

****** All numbers of the form 3{0}511 are divisible by 2, thus cannot be prime

*** If there are no 5's in {}, then the form is 3{0,1,6}1

**** If there are no 1's in {}, then the form is 3{0,6}1

***** All numbers of the form 3{0,6}1 are divisible by 2, thus cannot be prime

**** If there are 0's and 1's and 6's in {}, since 3101 and 3611 are primes, thus the 0's must before the 1's and the 1's must before the 6's

***** We have the prime [B]30161[/B]

**** If there no 6's in {}, then the form is 3{0,1}1

***** Since [B]3101[/B] is prime, we only need to consider the family 3{0}{1}1

****** Since [B]301111[/B] is prime, we only need to consider the families 3{1}1, 3{0}1, 3{0}11, 3{0}111

******* All numbers of the form 3{1}1 factored as (27*10^n-1)/8 = (5*3^n-1)/2 * (5*3^n+1)/4 (if n is odd) or (5*3^n-1)/4 * (5*3^n+1)/2 (if n is even), thus cannot be prime

******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime

******* The smallest prime of the form 3{0}11 is [B]300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011[/B] with 1158 0's, which can be written as 3(0^1158)11 and equal the prime 3*9^1160+10 ([URL="http://factordb.com/cert.php?id=1100000002376318423"]primality certificate of this prime[/URL])

******* All numbers of the form 3{0}111 are divisible by 2, thus cannot be prime

**** If there no 0's in {}, then the form is 3{1,6}1

***** Since [B]3611[/B] is prime, we only need to consider the family 3{1}{6}1

****** Since 661 is prime, we only need to consider the families 3{1}1 and 3{1}61

******* All numbers of the form 3{1}1 factored as (27*10^n-1)/8 = (5*3^n-1)/2 * (5*3^n+1)/4 (if n is odd) or (5*3^n-1)/4 * (5*3^n+1)/2 (if n is even), thus cannot be prime

******* The smallest prime of the form 3{1}61 is [B]311111111161[/B]

sweety439 2022-02-12 23:20

* Case (3,2):

** [B]32[/B] is prime, and thus the only minimal prime in this family.

* Case (3,4):

** [B]34[/B] is prime, and thus the only minimal prime in this family.

* Case (3,5):

** Since 32, 34, 25, 45, 65, [B]315[/B], [B]355[/B], [B]375[/B] are primes, we only need to consider the family 3{0,3,8}5 (since any digits 1, 2, 4, 5, 6, 7 between them will produce smaller primes)

*** If there is no 8 in {}, then the form is 3{0,3}5

**** If there is no 0 in {}, then the form is 3{3}5

***** All numbers of the form 3{3}5 are divisible by 2 (if the total number of 3's is odd) or 5 (if the total number of 3's is even), thus cannot be prime

**** If there is at least one 0 in {}, then there must be either <= 1 3's in {} before the rightmost 0 in {} or no 3's in {} after the rightmost 0 in {}, otherwise [B]333035[/B] will be a subsequence (in fact, not only for the rightmost 0 in {}, this is true for any 0 in {})

***** If there is no 3's in {} after the rightmost 0, then the form is 3{0,3}05

******

***** If there is no 3's in {} before the rightmost 0, then the form is 3{0}{3}5

****** Since [B]30333335[/B] is prime, we only need to consider the families 3{3}5, 3{0}35, 3{0}335, 3{0}3335, 3{0}33335

******* All numbers of the form 3{3}5 are divisible by 2 (if the total number of 3's is odd) or 5 (if the total number of 3's is even), thus cannot be prime

******* The smallest prime of the form 3{0}35 is [B]300000000035[/B]

******* All numbers of the form 3{0}335 are divisible by 2, thus cannot be prime

******* The smallest prime of the form 3{0}3335 is 30000000003335 (not minimal prime, since 300000000035 is prime)

******* All numbers of the form 3{0}33335 are divisible by 2, thus cannot be prime

***** If there is exactly one 3's in {} before the rightmost 0, then the form is 3{0}{3}05

****** Since 30333335 is prime, we only need to consider the families 3{3}05, 3{0}305, 3{0}3305, 3{0}33305, 3{0}333305

******* All numbers of the form 3{3}05 are divisible by 2 (if the total number of 3's is odd) or 5 (if the total number of 3's is even), thus cannot be prime

******* The smallest prime of the form 3{0}305 is 3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000305 (not minimal prime, since 300000000035 is prime)

******* All numbers of the form 3{0}3305 are divisible by 2, thus cannot be prime

******* The smallest prime of the form 3{0}33305 is [B]300033305[/B]

******* All numbers of the form 3{0}333305 are divisible by 2, thus cannot be prime

sweety439 2022-02-14 14:44

3x5 base 9

x={0,3}

minimal elements of x in {0,3} such that 3x5 (base 9) is prime:

[CODE]
3303
03033
030000
033333
0003000
0003330
0000000003
[/CODE]

x must contain an odd number of 3's

if x is of these forms, then 3x5 (base 9) cannot be prime:

[CODE]
{3}
{3}{0}
30{3}
[/CODE]

Note that 3x cannot be prime since it is always divisible by 3, and x5 is a subset of the original numbers (i.e. 3x5), since it must be begin with 3 (numbers cannot be [URL="https://en.wikipedia.org/wiki/Leading_zero"]begin with 0[/URL])

Edit: Found 3033300005 (base 9) is prime, but this is not minimal prime (start with b+1) since 30300005 (base 9) is prime.

Since the numbers of the form {3}{0}5 can be excluded as they are always divisible by either 2 or 5, "03" must be subsequence of x, thus ....

* The maximum number of 0's in x is 11, since
** There must be at least one 3 in x, or the number is of the form 3{0}5 and divisible by 2, thus let the form be 3y3z5 (i.e. let x be y3z), we can let the "3" in y3z be the rightmost 3 in x, i.e. there is no 3 in z
*** There are at most 8 0's in y, or 300000000035 will be a subsequence
*** There are at most 3 0's in z, or 30300005 will be a subsequence (note that there must be at least one 0 in y, or the form will be {3}{0}5 (note that we already assume that there is no 3 in z), and this form is already ruled out (as all numbers of this form are divisible by either 2 or 5))

* The maximum number of 3's in x is 5, since
** There must be at least one 0 in x, or the number is of the form 3{3}5 and divisible by either 2 or 5, thus let the form be 3y0z5 (i.e. let x be y0z), we can let the "0" in y0z be the leftmost 0 in x, i.e. there is no 0 in y
*** There are at most 1 3's in y, or 333035 will be a subsequence (note that there must be at least one 3 in z, or the form will be {3}{0}5 (note that we already assume that there is no 0 in y), and this form is already ruled out (as all numbers of this form are divisible by either 2 or 5))
*** There are at most 4 3's in z, or 30333335 will be a subsequence

Thus, the length of x is at most 16, and thus the length of the minimal prime (start with b+1) in base b=9, starting with 3 and ending with 5, is at most 18

sweety439 2022-02-25 11:42

4 Attachment(s)
update PRIMO certifate files of some primes in this project (B(0^1765)999B (base b=12) is not minimal prime (start with b+1), since B(0^27)9B (base b=12) is prime, but B(0^1765)999B (base b=12) still appear in the proof, currently the only unproven PRP appearing in the proof is 3(5^9234)4 (base b=7))

sweety439 2022-03-01 18:59

Proving that the set of the minimal primes (start with b+1) in base b is S is equivalent to:

* Proving that all strings in S are primes
* Proving that all strings in S are pairwise incomparable (for the subsequence ordering)
* Proving that all strings > b not contain any string in S as subsequence are composite

The first part needs to use [URL="http://www.ellipsa.eu/public/primo/primo.html"]Primo[/URL] to prove, for small primes it is very easy to prove, also for numbers < 10^1000 using [URL="https://www.numberempire.com/primenumbers.php"]this website[/URL] to prove or disprove (maybe they are [URL="https://primes.utm.edu/glossary/xpage/Pseudoprime.html"]pseudoprimes[/URL]), also for numbers whose N-1 and/or N+1 can be >=33.3333% factored, use [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 test[/URL] or [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL] to prove

The second part is just obvious, or if not obvious, you can use the "subsequence checker"

The third part needs to use number theory theorems to show that some families contain no primes (only count the numbers > b), and if S contains large primes, you should use [URL="https://en.wikipedia.org/wiki/Trial_division"]trial division[/URL] and [URL="https://en.wikipedia.org/wiki/Fermat_primality_test"]Fermat primality test[/URL] to show that these primes are the first prime (only count the numbers > b) in the corresponding families (by using the sieving program [URL="https://www.rieselprime.de/ziki/Srsieve"]srsieve[/URL] and the primality testing program [URL="https://www.rieselprime.de/ziki/PFGW"]PFGW[/URL] or [URL="https://www.rieselprime.de/ziki/LLR"]LLR[/URL])

sweety439 2022-03-01 19:19

e.g.

proving S(2) = {11}

* 11 (decimal 3) is prime (trivial, just use the [URL="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes"]sieve of Eratosthenes[/URL])
* all strings in {11} are pairwise incomparable (trivial, since the set {11} has only one element: 11)
* all strings > 10 (decimal 2) not contain any string in {11} as subsequence are composite (very easy to prove, since such string must contain at most one 1, and since if it contains no 1 then its value will be 0 and not > 10, it must contain exactly one 1, and since it is > 10, it must contain at least two 0's in the right of the 1, thus the number is divisible by 100 and hence composite)

proving S(3) = {12, 21, 111}

* 12 (decimal 5), 21 (decimal 7), 111 (decimal 13) are all primes (trivial, just use the [URL="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes"]sieve of Eratosthenes[/URL])
* all strings in {12, 21, 111} are pairwise incomparable (just obvious, none of 12, 21, 111 contains another of 12, 21, 111 as subsequence)
* all strings > 10 (decimal 3) not contain any string in {12, 21, 111} as subsequence are composite (since such numbers cannot contain 1 and 2 simultaneously (or either 12 or 21 will be a subsequence), and if it does not contain 1 then it contains only 0 and 2 (or only 0, or only 2), and hence its value must be even, and since it is > 10, it must be composite, besides, if it does not contain 2 then it contains only 0 and 1 (or only 0, or only 1), but it can contain at most two 1's (or 111 will be a subsequence), however, if it contain no 1 or exactly two 1, its value must be even, and since it is > 10, it must be composite, and if it contain exactly one 1, since it is > 10, it must contain at least two 0's in the right of the 1, thus the number is divisible by 100 and hence composite)

proving S(10) = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

* 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are all primes (you can use [URL="https://oeis.org/A000040/b000040_1.txt"]this online list[/URL] to check all primes in S up to 5200007, and for large primes you can either [URL="http://www.ellipsa.eu/public/primo/primo.html"]Primo[/URL] to prove or use [URL="https://www.numberempire.com/primenumbers.php"]this website[/URL] to check whether they are prime or not)
* all strings in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} are pairwise incomparable (just obvious, if not obvious for you, you can use the "subsequence checker" to check if they contain another string in S as subsequence or not)
* all strings > 10 not contain any string in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite (first, you need to use the divisibility rule of 2, 3, 5 to remove the numbers in some families, then prove that all numbers of the form 28{0}7 or 4{6}9 (as well as the numbers containing only 0 and 7) are divisible by 7, for some non-obvious numbers, like 6049 and 50027, you need to use trial division or Fermat primality test to show that they are composite)

sweety439 2022-03-05 21:17

1 Attachment(s)
Upload newest pdf file about this research.

sweety439 2022-03-11 21:41

Like the sense of [URL="https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122"]https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122[/URL] and [URL="https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7"]https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7[/URL], for the minimal prime (start with b+1) problem base b:

* Bases 2, 3, 4 have no minimal primes (start with b+1) with >3 digits (see [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]this article[/URL]), thus the "unsolved family" list for these bases will be empty.

* For base 5, "unsolved family" list should include the non-simple families 1{0}1{0}3 and 3{0}3{0}1, and with primes 10103, 30301, 33001, these families become 1{0}13, 3{0}31 (note: 11{0}3 can be removed, since all numbers in this family are divisible by 3)

* For base 5, "unsolved family" list should include families 1{4}, {3}1, {4}1

* For base 10, "unsolved family" list should include the non-simple families 2{0}2{0}1 and 5{0}2{0}7 and 6{0,6}49, and with primes 20201, 20021, 50207, 60649, these families become 22{0}1, 52{0}7, 5{0}27, {6}{0}49, and with the prime 666649, the family {6}{0}49 becomes 6{0}49, 66{0}49, 666{0}49

* For base 10, "unsolved family" list should include families {5}1, 8{5}1, 80{5}1, and with the prime 80555551, the family 8{5}1 become unneeded and removed, and only leave the family {5}1

sweety439 2022-03-25 19:12

[URL="https://oeis.org/A034388"]https://oeis.org/A034388[/URL]: Smallest prime containing at least n consecutive identical digits.

This sequence is related the project in this forum, since for example, the largest minimal prime (start with b+1) in base b=10 is 5(0^28)27, it makes that [URL="https://oeis.org/A034388"]A034388[/URL](28) <= 5(0^28)27

Similarly, for base b=16, D(B^32234) is minimal prime (start with b+1), assuming its primality, thus for the analog sequence of [URL="https://oeis.org/A034388"]https://oeis.org/A034388[/URL] in base 16, a(32234) <= D(B^32234), also for base b=14, 4(D^19698) is minimal prime (start with b+1), thus for the analog sequence of [URL="https://oeis.org/A034388"]https://oeis.org/A034388[/URL] in base 14, a(19698) <= 4(D^19698), and for base b=13, 8(0^32017)111 is minimal prime (start with b+1), assuming its primality, thus for the analog sequence of [URL="https://oeis.org/A034388"]https://oeis.org/A034388[/URL] in base 13, a(32017) <= 8(0^32017)111

sweety439 2022-04-02 20:00

[QUOTE=sweety439;599662]In base b=8 such prime does not exist, since the family 1{0}1 has infinite subset whose elements are pairwise coprime but the family 1{0}1 can be ruled out as only contain composite numbers, but if we only consider the numbers not in the family 1{0}1, such prime will be the smallest prime factor of (4*8^217+17)/7 (if it is > 7885303569123738614221) or 7885303569123738614221, which is needed to remove the composites (4^216)7 and (4^116)7, respectively, see [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {4}7 in base 8[/URL][/QUOTE]

(4*8^217+17)/7 has been factored by me (using [URL="http://www.alpertron.com.ar/ECM.HTM"]ECM Integer factorization calculator[/URL]), one prime factor 46096827569809626166519 was found, and the remaining number is a 174-digit composite.

Since 46096827569809626166519 (23 digits) > 7885303569123738614221 (22 digits), if we only consider the numbers not in the family 1{0}1, such prime is 46096827569809626166519 if [URL="http://factordb.com/index.php?id=1100000002999513593"]the 174-digit composite[/URL] has no prime factor < 46096827569809626166519, I just use the ECM factorization calculator to find this 23-digit factor, this factor may not be the smallest prime factor of (4*8^217+17)/7, like the status of the Mersenne number [URL="https://www.mersenne.org/report_exponent/?exp_lo=1237&full=1"]M1237[/URL], it is unknown whether the only known 70-digit prime factor is the smallest prime factor of M1237, see [URL="https://oeis.org/A016047"]https://oeis.org/A016047[/URL], if there is a smaller prime factor of (4*8^217+17)/7, then....

* If this prime factor is < 7885303569123738614221, then such prime in base 8 is 7885303569123738614221
* If this prime factor is between 7885303569123738614221 and 46096827569809626166519, then such prime in base 8 is this prime

See [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of (4*8^(n+1)+17)/7 in factordb[/URL], since for odd n this number is divisible by 3 (and 3 must be the smallest prime factor of this number), we only list even n

sweety439 2022-04-05 07:59

The smallest prime of the form y{z} in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b):

3, 5, 11, 19, 29, 41, 3583, 71, 89, 109, 131, 2027, 181, 408700964355468749, 239, 271, 5507, 846825857, 379, 419, 461, 17276416353328819798072137388863592892072278184923153720493777138850572564953, 839967991029301247, 599, 3885038158778096269468893991882380063764065770433606110283149695964997245520484669311748838825973451239771955518933348332721403496018696846203290707966794803507099534240007184258836096614399, 701, 2368778164222232774191928573951, 811, 26099, 929, 991, 34847, 3095263992211830248865791, 1457749, 1259

The smallest prime of the form z{0}1 in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b):

3, 7, 13, 101, 31, 43, 449, 73, 9001, 259374246011, 19009, 157, 2549, 211, 241, 1336337, 307, 218336795902605993201009018384568383223, 31129600000000000001, 421, 463, 255042399139852495799, 13249, 601, 16901, 13817467, 757, 23549, 23490001, 858874531, 35740566642812256257, 34849, 1123, 41651, 45361

The smallest prime of the form {z}1 in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b):

3, 7, 13, 3121, 31, 43, 549755813881, 73, 991, 1321, 248821, 157, 2731, 211, 241, 34271896307617, 307, 6841, 13107199999999999999981, 421, 463, 141050039560662968926081, 331753, 601, 17551, 7625597484961, 757, 1816075630094014572464024421543167816955354437761, 21869999971, 29761, 34359738337, 1185889, 1123, 42841, 60466141

The smallest prime of the form 1{0}z in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b):

3, 5, 7, 29, 11, 13, 71, 17, 19, 131, 23, 181, 2177953337809371149, 29, 31, 83537, 5849, 37, 419, 41, 43, 279863, 47, 15649, 701, 53, 811, 420707233300229, 59, 61, 3599131035634557106248430806148785487095757694641533306480604458089470064537190296255232548883112685719936728506816716098566612844395439751206812144692131084107807, 35969, 67, 1259, 71

The smallest prime of the form {y}z in base b for b = 3, 4, 5, ..., 36 are (always minimal prime (start with b+1) base b): (this form is not interpretable in base 2)

5, 11, 19, 29, 41, 439, 71, 89, 109, 131, 16836900297891418080414469547118518955584357920776290786511507224819852347973193037600665289070901330976115445902783343792856149076064327963454445124840887022352433623214149015015943271257627167012185236811023315748308075343126054090560004563875124190448995227748073744916159908957819701603274854998000296763254125672206384758348891742961717040363229489213108521955314350073857925001010097317113705164622416602981584525394558649693204742511309000575073486313783914987497483013408328355077527202814535784777000148396721007194688339582681878366906510944731328876064735814127172451578146421749559114747412555063799277435883965467381, 181, 535461077009, 239, 271, 98801, 754617425461612781, 379, 419, 461, 74751395041, 317351, 599, 11406121, 701, 21139, 811, 703862069, 929, 991, 44846087994920604803621301910895025159221288572184381044859909289485641865275583407994915558605570950946734680988452965394713688106491174400948294112101434213220953385460836118066048540254897963702856420719776463509047941852241566428193327643900875942259940601970070393196751457213209881319085660701362499099654878864165624569792466096820413058979402445330988995938988157868541715781, 38113, 7385772222129586256251615636489, 1259

sweety439 2022-04-05 08:08

The smallest prime of the form {z}y in base b for b = 3, 4, 5, ..., 36 are (always minimal prime (start with b+1) base b): (0 if no such prime exists) (this form is not interpretable in base 2)

7, 0, 23, 0, 47, 0, 79, 0, 14639, 0, 167, 0, 223, 0, 24137567, 0, 359, 0, 439, 0, 480250763996501976790165756943039, 0, 6103515623, 0, 727, 0, 839, 0, 29789, 0, 1087, 0, 1223, 0

The smallest prime of the form 1{0}2 in base b for b = 3, 4, 5, ..., 36 are (always minimal prime (start with b+1) base b): (0 if no such prime exists) (this form is not interpretable in base 2)

5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 0, 0, 23, 0, 952809757913929, 0, 0, 0, 29, 0, 31, 0, 0, 0, 1091, 0, 37, 0

The smallest prime of the form 1{z} in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b):

3, 5, 7, 1249, 11, 13, 127, 17, 19, 241, 23, 337, 76831, 29, 31, 577, 647, 37, 20479999999999, 41, 43, 296071777, 47, 1249, 617831551, 53, 1567, 15387133080032326246081223292828787411221911122916017220126284227825703776392672467768318856009763825207593900596158761682711294895921233392537083406917227083982402321012446032594528728383203531755841, 59, 61, 2147483647, 2582935937, 67, 3676531249, 71

The smallest prime of the form 2{0}1 in base b for b = 3, 4, 5, ..., 36 are (always minimal prime (start with b+1) base b): (0 if no such prime exists) (this form is not interpretable in base 2)

7, 0, 11, 13, 0, 17, 19, 0, 23, 3457, 0, 29, 31, 0, 13555929465559461990942712143872578804076607708197374744547, 37, 0, 41, 43, 0, 47, 1153, 0, 53, 1459, 0, 59, 61, 0, 65537, 67, 0, 71, 73

The smallest prime of the form 1{0}1 in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b): (0 if no such prime exists)

3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 197, 0, 17, 0, 19, 0, 401, 0, 23, 0, 577, 0, 677, 0, 29, 0, 31, 0, 0, 0, 1336337, 0, 37

The smallest prime of the form {1} in base b for b = 2, 3, 4, ..., 36 are (always minimal prime (start with b+1) base b): (0 if no such prime exists)

3, 13, 5, 31, 7, 2801, 73, 0, 11, 50544702849929377, 13, 30941, 211, 241, 17, 307, 19, 109912203092239643840221, 421, 463, 23, 292561, 601, 0, 321272407, 757, 29, 732541, 31, 917087137, 0, 1123, 2458736461986831391, 57785511861854089559605684285384747472873075954938549266821996762520614682090417010479587236790517340193840109863642510356045237096340500854836834673594590986502765133399405931445515950293723048093118292954035082630781507315268041070570042738804650015484793905221070413101021864355439951875266340353210153398276807146377561258956649022201316646128234211457693681312361704211065831222237374054447781785197765525068555496240581389620280398439369732560881984414556748507653965669519761, 37

sweety439 2022-04-05 08:26

[QUOTE=sweety439;603131](4*8^217+17)/7 has been factored by me (using [URL="http://www.alpertron.com.ar/ECM.HTM"]ECM Integer factorization calculator[/URL]), one prime factor 46096827569809626166519 was found, and the remaining number is a 174-digit composite.

Since 46096827569809626166519 (23 digits) > 7885303569123738614221 (22 digits), if we only consider the numbers not in the family 1{0}1, such prime is 46096827569809626166519 if [URL="http://factordb.com/index.php?id=1100000002999513593"]the 174-digit composite[/URL] has no prime factor < 46096827569809626166519, I just use the ECM factorization calculator to find this 23-digit factor, this factor may not be the smallest prime factor of (4*8^217+17)/7, like the status of the Mersenne number [URL="https://www.mersenne.org/report_exponent/?exp_lo=1237&full=1"]M1237[/URL], it is unknown whether the only known 70-digit prime factor is the smallest prime factor of M1237, see [URL="https://oeis.org/A016047"]https://oeis.org/A016047[/URL], if there is a smaller prime factor of (4*8^217+17)/7, then....

* If this prime factor is < 7885303569123738614221, then such prime in base 8 is 7885303569123738614221
* If this prime factor is between 7885303569123738614221 and 46096827569809626166519, then such prime in base 8 is this prime

See [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of (4*8^(n+1)+17)/7 in factordb[/URL], since for odd n this number is divisible by 3 (and 3 must be the smallest prime factor of this number), we only list even n[/QUOTE]

(4*8^217+17)/7 has been fully factored: [URL="http://factordb.com/index.php?id=1100000000530031351"]factordb entry[/URL], its smallest prime factor is surely 46096827569809626166519, and since the smallest prime factor of (4*8^(n+1)+17)/7 is < 46096827569809626166519 for all other n < 220 (see factorization of (4*8^(n+1)+17)/7 for [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]1<=n<=200[/URL] [URL="http://factordb.com/index.php?query=%284%2A8%5E%28n%2B1%29%2B17%29%2F7&use=n&perpage=200&format=1&sent=1&PR=1&PRP=1&C=1&CF=1&U=1&FF=1&VP=1&EV=1&OD=1&VC=1&n=201"]201<=n<=400[/URL]) (the smallest prime of the form (4*8^(n+1)+17)/7 is n = 220), thus for base 8, this prime in post [URL="https://mersenneforum.org/showpost.php?p=599662&postcount=296"]#296[/URL] is surely 46096827569809626166519 if we do not count the numbers of the form 1{0}1 (since the family 1{0}1 has infinite subset whose elements are pairwise coprime but the family 1{0}1 can be ruled out as only contain composite numbers, thus such prime does not exist if we count all numbers of the form 1{0}1), however, all base 8 strings with length <= 1048576 not containing any element in S (S is the set of the minimal primes (start with b+1) in base b = 8) as subsequence, when read as base b = 8 representation, which are > b, are divisible by at least one prime <= 46096827569809626166519, since the smallest number of the form 1{0}1 in base 8 which does not have a prime factor <= 46096827569809626166519 is 8^(2^20)+1, reference: [URL="http://factordb.com/index.php?query=8^(2^n)%2B1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 8^(2^n)+1 in factordb[/URL] [URL="http://www.prothsearch.com/fermat.html"]factorization of Fermat numbers 2^(2^n)+1[/URL] [URL="http://www.prothsearch.com/GFNsmall.html"]Factorization of generalized Fermat numbers for small indices n, including 2^(2^n)+1 and (8^(2^n)+1)/(2^(2^n)+1)[/URL] (note that 8^n+1 has a smaller factor 8^m+1 if n is not power of 2)

[CODE]
(4^216)7 (base 8) = (4*8^217+17)/7 = 5339359077080146175872639684842398076217903990197171850186392158090123167110497878608062535758843786712376828575269340669880267011072512583097971647309196947165762910337041495868415054697739733287 = 46096827569809626166519 * 11488335888396928554793823 * 6752215872441605227672501208714569986307360549 * 1493188537980759209087329477589240729409871102777007948247089531288003768186691760462059984190495270499
[/CODE]

sweety439 2022-04-05 08:32

List of primes of given forms:

1{0}1:

[URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL]
[URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]

{1}:

[URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL]
[URL="https://archive.ph/tf7jx"]https://archive.ph/tf7jx[/URL]

y{z}, z{0}1, 10{z}, 11{0}1, {z}1, 1{0}z, {z}yz, 1{0}11:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]

y{z}:

[URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL]

1{z}, 2{0}1, {z}y, 1{0}2:

(I am now computing ....)

{z0}z1:

[URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf[/URL]

{#}$:

[URL="https://oeis.org/A253242/a253242.txt"]https://oeis.org/A253242/a253242.txt[/URL]
[URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL] (sorted by n instead of by b)

sweety439 2022-04-06 09:31

[QUOTE=sweety439;597528]Proving the set of the minimal primes (start with b+1) in base b is S, is equivalent to:

* Prove that all elements in S, when read as base b representation, are primes > b.

* Prove that all proper subsequence of all elements in S, when read as base b representation, which are > b, are composite.

* Prove that all primes > b, when written in base b, contain at least one element in S as subsequence (equivalently, prove that all strings not containing any element in S as subsequence, when read as base b representation, which are > b, are composite).[/QUOTE]

In fact, the third part covers the second part, since all proper subsequence of all elements in S do not containing any element in S as subsequence, thus if a such subsequence is prime > b, it will also be counterexample of the third part.

sweety439 2022-04-06 09:40

A Prime Game
 
Write down a prime number > 10, and I can always strike out 0 or more digits to get a prime in this list:

[CODE]
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
227
251
257
277
281
349
409
449
499
521
557
577
587
727
757
787
821
827
857
877
881
887
991
2087
2221
5051
5081
5501
5581
5801
5851
6469
6949
8501
9001
9049
9221
9551
9649
9851
9949
20021
20201
50207
60649
80051
666649
946669
5200007
22000001
60000049
66000049
66600049
80555551
555555555551
5000000000000000000000000000027
[/CODE]

(I have a proof for this)

Dobri 2022-04-06 10:50

2[SUP]82589933[/SUP]-1

kruoli 2022-04-06 11:12

Since 2[SUP]82,589,933[/SUP] is 1…1 in decimal, it can be reduced to 11.

Dobri 2022-04-06 11:58

[I]M[/I][SUB]52[/SUB]

axn 2022-04-06 12:12

[QUOTE=kruoli;603377]Since 2[SUP]82,589,933[/SUP] is 1…1 in [B]decimal[/B][/QUOTE]

Are you sure?

Dr Sardonicus 2022-04-06 12:16

[QUOTE=kruoli;603377]Since 2[SUP]82,589,933[/SUP] is 1…1 in decimal, it can be reduced to 11.[/QUOTE]:shock:

It's 1...1 in [i]binary[/i], not decimal. In decimal, 2[sup]82589933[/sup] - 1 has the 3-digit terminus 591 which contains the string 59, which is on the list.

kruoli 2022-04-06 12:26

The official decimal expansion of it on the mersenne.org website shows that it is 1…1 both in binary and in decimal! But in decimal, it is not 11…11, while it is (of course) in binary.

Edit, clarification: By 1…1, I do not mean only ones. It should be understood as starting with a one, other digits, and ending with a one.

Dr Sardonicus 2022-04-06 14:06

[QUOTE=kruoli;603388]<snip>
Edit, clarification: By 1…1, I do not mean only ones. It should be understood as starting with a one, other digits, and ending with a one.[/QUOTE]For possible future reference, giving the first two (or more) most significant digits and the last two (or more) digits of a many-digit number is common practice. Sometimes the number of intervening digits is also given.

In the present instance, 14...91 or 14... (24862044 digits) ...91 would be much clearer.

The most significant digits of M[sub]p[/sub] are determined by the fractional part of p*log(2)/log(10).

[b]EDIT:[/b] Although there is no upper bound on the number of leading 1's in a Mersenne number 2[sup]n[/sup] - 1, or AFAIK for 2[sup]p[/sup] - 1 with p prime, there [i]is[/i] an upper bound for the number of terminal 1's. Three. The proof is easy.

(The Mersenne number 2[sup]888689[/sup] - 1 has 4 leading 1's and 3 terminal 1's. M[sub]888689[/sub] has a small factor q = 24*p + 1, and the cofactor is proven composite.)

sweety439 2022-04-07 03:12

[QUOTE=sweety439;599662]An interesting and difficult problem is find the smallest prime p such that all strings not containing any element in S (S is the set of the minimal primes (start with b+1) in base b) as subsequence, when read as base b representation, which are > b, are divisible by at least one prime <= p (such prime may not exist, as in some bases (such as 8, 9, 12, 14, 16), there are some families which are ruled out as only contain composite numbers by all or partial algebraic factors, in these bases we find the smallest prime p such that all strings not containing any element in S (S is the set of the minimal primes (start with b+1) in base b) as subsequence, when read as base b representation, which are > b, are either "divisible by at least one prime <= p" or "of the special form", the special forms are:

[CODE]
base (b) forms
8 1{0}1
9 {1}, 3{1}, 3{8}, 3{8}35, {8}5
12 {B}9B
14 8{D}, {D}5
16 10{5}, 1{5}, {4}1, 7{3}, 8{5}, 8{F}, B{4}1, {C}D, {C}DD, {F}7
17 1{9}, {9}8
19 1{6}, {6}5, 7{2}, 89{6}
24 3{N}, 5{N}, {6}1, 8{N}
25 {1}, 1{3}, 1{8}, 2{1}, {3}2, 5{1}, 5{8}, 7{1}, {8}3, {8}7, C{1}, F{1}, M{1}, 27{1}
27 1{0}8, 7{Q}, 8{0}1, 9{G}, {D}E, {G}7, {Q}J
32 1{0}1, {1}
33 F{W}, L{4}, {W}H
34 1{B}, 8{X}, G{B}, L{B}, {X}P
36 3{7}, 3{Z}, 8{Z}, 9{5}, G{7}, O{Z}, {Z}B
[/CODE]

The sequence of the smallest prime factor of the numbers in these families is very likely to be unbounded above, thus such p in these bases very unlikely to exist.

Equivalently, finding the smallest prime p such that "the set of the minimal ("the primes > b and <= p" and "the numbers > b not divisible by any prime <= p") in base b" is the same as "the set of the minimal primes (start with b+1) in base b"

In base b=10 such prime is 214741, which is needed to remove the composite number 5(0^24)27, see [URL="http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 5{0}27[/URL] and [URL="http://factordb.com/index.php?query=%285*10%5E%28n%2B1%29-41%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {5}1[/URL]

In base b=2 such prime is 10 (decimal 2), since all numbers > 1 not divisible by 2 (i.e. the odd numbers > 1) have first digit 1 and last digit 1 in base b=2, thus the only minimal prime (start with b+1) 11 (decimal 3) must be a subsequence.

In base b=3 such prime is 10 (decimal 3); first, 10 (decimal 3) is needed, since this prime is needed to remove the numbers 1{0} (i.e. 100, 1000, 10000, 100000, ...); second, for the numbers not divisible by 2 or 3, such number cannot end with 0 (of course also cannot begin with 0) and must have an odd number of 1, thus either 111 (i.e. 3 1's) or 12 or 21 must be a subsequence, unless the number is 1, which is not allowed in this research.

In base b=4 such prime is 3 (decimal 3), which is used to remove the numbers: 21 (decimal 9), all numbers of the form 2{0}1, and the numbers containing only 0 and 3

In base b=5 such prime is 11332432 (decimal 105367), which is used to remove the number 1(0^53)13, see [URL="http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 1{0}13 in base 5[/URL]

In base b=6 such prime is 21 (decimal 13), which is used to remove the number 441 (decimal 169), note that 4041 (decimal 889) is divisible by 7

In base b=7 such prime is 15421 (decimal 4327), see [URL="http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {3}1 in base 7[/URL] and [URL="http://factordb.com/index.php?query=36*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 51{0}1 in base 7[/URL]

In base b=8 such prime does not exist, since the family 1{0}1 has infinite subset whose elements are pairwise coprime but the family 1{0}1 can be ruled out as only contain composite numbers, but if we only consider the numbers not in the family 1{0}1, such prime will be the smallest prime factor of (4*8^217+17)/7 (if it is > 7885303569123738614221) or 7885303569123738614221, which is needed to remove the composites (4^216)7 and (4^116)7, respectively, see [URL="http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of {4}7 in base 8[/URL]

In base b=9 such prime does not exist, since the families {1}, 3{1}, 3{8}, 3{8}35, {8}5 has infinite subset whose elements are pairwise coprime but the families {1}, 3{1}, 3{8}, 3{8}35, {8}5 can be ruled out as only contain composite numbers, but if we only consider the numbers not in the families {1}, 3{1}, 3{8}, 3{8}35, {8}5, such prime exists, but is very hard to find, since finding this prime requires factoring the large numbers in the families 3{0}11, 2{7}07, 7{6}2

In base b=12 such prime does not exist, since the family {B}9B has infinite subset whose elements are pairwise coprime but the family {B}9B can be ruled out as only contain composite numbers, but if we only consider the numbers not in the family {B}9B, such prime will be 534AB547A0351 (decimal 47113717465069), see [URL="http://factordb.com/index.php?query=4*12%5E%28n%2B2%29%2B91&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 4{0}77 in base 12[/URL] and [URL="http://factordb.com/index.php?query=11*12%5E%28n%2B2%29%2B119&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of B{0}9B in base 12[/URL][/QUOTE]

A highly-related problem is find the smallest set T (the set with the smallest "number of elements", then with the smallest "smallest prime", then with the smallest "second-smallest prime", then with the smallest "third-smallest prime", ...) of primes such that all strings not containing any element in S (S is the set of the minimal primes (start with b+1) in base b) as subsequence, when read as base b representation, which are > b, are divisible by at least one element in T (in some bases (such as 8, 9, 12, 14, 16), there are some families which are ruled out as only contain composite numbers by all or partial algebraic factors, thus in these bases, the set T is infinite)

In base b=2, T={2}, since all numbers > 1 not divisible by 2 (i.e. the odd numbers > 1) have first digit 1 and last digit 1 in base b=2, thus the only minimal prime (start with b+1) 11 (decimal 3) must be a subsequence.

In base b=3, T={2,3}; first, 10 (decimal 3) is needed, since this prime is needed to remove the numbers 1{0} (i.e. 100, 1000, 10000, 100000, ...); second, for the numbers not divisible by 2 or 3, such number cannot end with 0 (of course also cannot begin with 0) and must have an odd number of 1, thus either 111 (i.e. 3 1's) or 12 or 21 must be a subsequence, unless the number is 1, which is not allowed in this research.

In base b=4, T={2,3}; since all numbers not divisible by 2 (i.e. end with 1 or 3) not contain any of {11, 13, 23, 31, 221} as subsequence are either of the form {0}2{0}1 or contain only 0 and 3, thus must be divisible by 3

In base b=5, T={2,3,5,7,11,13,17,23,31,41,47,251,691,887,5081,7219,82609,105367}, see [URL="http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]factorization of 1{0}13 in base 5[/URL]

[CODE]
prime use to remove
2 (2) 1{0,4}1, 1{0}3, 11{4}4, 2{0,2,4}2, 2{0,2,4}4, 30{0}01, 33{0}31, 4{4}11, 4{0,2,4}2, 4{0,2,4}4
3 (3) 11{0}3, 3{0,3}11, 3{0,3}3, 1(0^n)13 for all even n (including n=0) and n < 93
10 (5) all numbers ending with 0
12 (7) 144, 331, 1(0^n)13 for all n == 1 mod 6 and n < 93
21 (11) 3301, 441, 1(0^n)13 for all n divisible by 5 and n < 93
23 (13) 1(0^n)13 for all n == 3 mod 4 and n < 93
32 (17) 3031
43 (23) 1(0^81)13
111 (31) 30031
131 (41) 1(0^17)13, 1(0^57)13, 1(0^77)13
142 (47) 1(0^29)13
2001 (251) 1(0^89)13
10231 (691) 1(0^9)13
12022 (887) 1(0^21)13
130311 (5081) 1(0^41)13
212334 (7219) 1(0^69)13
10120414 (82609) 1(0^33)13
11332432 (105367) 1(0^53)13
[/CODE]

In base b=6, T={2,3,5,7,13}; since all numbers not divisible by 2 or 3 (i.e. end with 1 or 5) not contain any of {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041} as subsequence (except 441 (divisible by 13) and 4041 (divisible by 7)) are either of the form {0}4{0}1 or contain only 0 and 5, thus must be divisible by 5

sweety439 2022-04-07 03:25

Since finding the set T requiring finding the set M(Lb) first, finding T is much harder than finding M(Lb)

For b = 8 and b = 9 and b = 12, T contains infinitely many numbers:

* For b = 8, T contains the smallest prime factor of 8^(2^n)+1 for all n
* For b = 9, T contains the smallest prime factor of (9^n-1)/8 for all prime n, also contain the smallest prime factor of (25*9^n-1)/8, 4*9^n-49, 4*9^n-1, 9^n-4 for all n
* For b = 12, T contains the smallest prime factor of 12^n-25 for all even n (for all odd n, the smallest prime factor is 13)

Theorem: For all base b, T must contain all primes <= b

Proof: For primes p < b, consider the combination of the digits divisible by p (including 0), and for prime p = b, consider the family 1{0}

We find the T for b = 10:

* T must contain {2,3,5,7}
* The smallest prime of the form 2{0}21 is 20021 --> 221 (=13*17) and 2021 (=43*47) must be removed, thus either 13 or 17 is needed, and either 43 or 47 is needed
* The smallest prime of the form 22{0}1 is 22000001 --> 221, 2201, 22001, 220001, 2200001 must be removed, thus the primes {13,31,19,73} are needed (7 is already in the set)
* The smallest prime of the form 5{5}1 is 555555555551 --> All smaller numbers in this family must be removed
* 581 must be removed, but 581 is divisible by 7 and 7 is already in the set
* The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 --> All smaller numbers in this family must be removed
* The smallest prime of the form 52{0}7 is 5200007 --> 527, 5207, 52007, 520007 must be removed, thus the primes {17,41,131,23} are needed
*

sweety439 2022-04-08 11:44

Factorization of x{y}z families other than Cunningham numbers:

[URL="https://stdkmd.net/nrr/#factortables"]https://stdkmd.net/nrr/#factortables[/URL]
[URL="http://mklasson.com/factors/index.php"]http://mklasson.com/factors/index.php[/URL]
[URL="https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt"]https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt[/URL]
[URL="https://oeis.org/A162491"]https://oeis.org/A162491[/URL]
[URL="https://oeis.org/A093810"]https://oeis.org/A093810[/URL]
[URL="https://oeis.org/A093817"]https://oeis.org/A093817[/URL]
[URL="https://oeis.org/A080443"]https://oeis.org/A080443[/URL]
[URL="https://oeis.org/A081715"]https://oeis.org/A081715[/URL]
[URL="https://oeis.org/A080798"]https://oeis.org/A080798[/URL]
[URL="https://oeis.org/A080892"]https://oeis.org/A080892[/URL]

sweety439 2022-04-11 02:13

2 Attachment(s)
[QUOTE=sweety439;603317]List of primes of given forms:

1{z}, 2{0}1, {z}y, 1{0}2:

(I am now computing ....)[/QUOTE]

for k = 2:

* A = numbers n such that 2*b^n-1 is prime --> family 1{z}
* B = numbers n such that (2*b^n+1)/gcd(b-1,3) is prime --> for b not == 1 mod 3, family 2{0}1 (for b == 1 mod 3, family 2{0}1 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3)
* C = numbers n such that (b^n-2)/gcd(b,2) is prime --> for odd b, family {z}y (for even b, family {z}y can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2)
* D = numbers n such that (b^n+2)/gcd(b,2)/gcd(b-1,3) is prime --> for b == 3 or 5 mod 6, family 1{0}2 (for b == 1 mod 3, family 1{0}2 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3, also, for even b, family 1{0}2 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2)

for k = 3:

* A = numbers n such that (3*b^n-1)/gcd(b-1,2) is prime --> for even b, family 2{z} (for odd b, family 2{z} can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2)
* B = numbers n such that (3*b^n+1)/gcd(b-1,4) is prime --> for even b, family 3{0}1 (for odd b, family 3{0}1 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2)
* C = numbers n such that (b^n-3)/gcd(b,3)/gcd(b-1,2) is prime --> for b == 2 or 4 mod 6, family {z}x (for odd b, family {z}x can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2, also, for b divisible by 3, family {z}x can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3)
* D = numbers n such that (b^n+3)/gcd(b,3)/gcd(b-1,4) is prime --> for b == 2 or 4 mod 6, family 1{0}3 (for odd b, family 1{0}3 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 2, also, for b divisible by 3, family 1{0}3 can be ruled out as only contain composite numbers, as all numbers in this family are divisible by 3)

sweety439 2022-04-11 02:39

We can expect the length of the largest minimal prime (start with b+1) in base b, and expect the number of unsolved families in base b when searched to certain limit (e.g. 10000 digits or 50000 digits), like [URL="http://chesswanks.com/num/LTPs/"]expectations of the length of the largest left truncatable primes in base b[/URL] and [URL="http://fatphil.org/maths/RightTruncatablePrimes.html"]expectations of the length of the largest right truncatable primes in base b[/URL]

There are examples of expectations related to this: [URL="https://mersenneforum.org/showpost.php?p=236438&postcount=22"]https://mersenneforum.org/showpost.php?p=236438&postcount=22[/URL] [URL="https://mersenneforum.org/showpost.php?p=236440&postcount=20"]https://mersenneforum.org/showpost.php?p=236440&postcount=20[/URL] [URL="https://mersenneforum.org/showpost.php?p=461665&postcount=7"]https://mersenneforum.org/showpost.php?p=461665&postcount=7[/URL] [URL="https://mersenneforum.org/showthread.php?t=26228"]https://mersenneforum.org/showthread.php?t=26228[/URL] [URL="https://oeis.org/A055557/a055557.txt"]https://oeis.org/A055557/a055557.txt[/URL] [URL="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906"]https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906[/URL] [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL] [URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]

sweety439 2022-04-21 18:25

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Upload newest pdf file.

sweety439 2022-05-07 12:39

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upload the base 9 file.

sweety439 2022-05-08 14:14

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bases 9, 10, 12

sweety439 2022-05-12 12:59

1 Attachment(s)
The possible length of a prime in the families (in any base b) are: (the length is possible if and only if the corresponding polynomial of the base b satisfies the condition in [URL="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture"]Bunyakovsky conjecture[/URL], and if Bunyakovsky conjecture is true, then there are infinitely many bases b such that the number with these lengths in the families are primes)

1{0}1: length is of the form 2^r+1 (b^n+1 is divisible by b^k+1 if k divides n and n/k is odd > 1, and b^n+1 satisfies both conditions in Bunyakovsky conjecture if n is of the form 2^r)
1{0}2: any length >= 2 (b^n+2 always satisfies both conditions in Bunyakovsky conjecture)
1{0}3: any length >= 2 (b^n+3 always satisfies both conditions in Bunyakovsky conjecture)
1{0}4: length not == 1 mod 4 (b^n+4 = (b^(n/2)-2*b^(n/4)+2) * (b^(n/2)+2*b^(n/4)+2) if n == 0 mod 4, and b^n+4 satisfies both conditions in Bunyakovsky conjecture if n not == 0 mod 4)
1{0}z: length not == 0 mod 6 (b^n+(b-1) has factor b^2-b+1 (z1 in base b) if n == 5 mod 6, and b^n+(b-1) satisfies both conditions in Bunyakovsky conjecture if n not == 5 mod 6)
{1}: length is prime ((b^n-1)/(b-1) is divisible by (b^k-1)/(b-1) if k divides n and k > 1 and n/k > 1 (for k = 1, (b^k-1)/(b-1) = 1), and (b^n-1)/(b-1) satisfies both conditions in Bunyakovsky conjecture if n is prime)
1{z}: any length >= 2 (2*b^n-1 always satisfies both conditions in Bunyakovsky conjecture)
2{0}1: any length >= 2 (2*b^n+1 always satisfies both conditions in Bunyakovsky conjecture)
2{z}: any length >= 2 (3*b^n-1 always satisfies both conditions in Bunyakovsky conjecture)
3{0}1: any length >= 2 (3*b^n+1 always satisfies both conditions in Bunyakovsky conjecture)
3{z}: length == 0 mod 2 (4*b^n-1 = (2*b^(n/2)-1) * (2*b^(n/2)+1) if n == 0 mod 2, and 4*b^n-1 satisfies both conditions in Bunyakovsky conjecture if n == 1 mod 2)
4{0}1: length not == 1 mod 4 (4*b^n+1 = (2*b^(n/2)-2*b^(n/4)+1) * (2*b^(n/2)+2*b^(n/4)+1) if n == 0 mod 4, and 4*b^n+1 satisfies both conditions in Bunyakovsky conjecture if n not == 0 mod 4)
{y}z: any length >= 2 (((b-2)*b^n+1)/(b-1) always satisfies both conditions in Bunyakovsky conjecture)
y{z}: length not == 5 mod 6 ((b-1)*b^n-1 has factor b^2-b+1 (z1 in base b) if n == 4 mod 6, and (b-1)*b^n-1 satisfies both conditions in Bunyakovsky conjecture if n not == 4 mod 6)
z{0}1: length 2 and length not == 2 mod 6 ((b-1)*b^n+1 has factor b^2-b+1 (z1 in base b) if n == 1 mod 6 and n > 1 (for n = 1, (b-1)*b^n+1 is exactly b^2-b+1 (z1 in base b) itself), and (b-1)*b^n+1 satisfies both conditions in Bunyakovsky conjecture if n not == 1 mod 6)
{z}1: length 2 and length not == 2 mod 6 (b^n-(b-1) has factor b^2-b+1 (z1 in base b) if n == 2 mod 6 and n > 2 (for n = 2, b^n-(b-1) is exactly b^2-b+1 (z1 in base b) itself), and b^n-(b-1) satisfies both conditions in Bunyakovsky conjecture if n not == 2 mod 6)
{z}w: length == 1 mod 2 (b^n-4 = (b^(n/2)-2) * (b^(n/2)+2) if n == 0 mod 2, and b^n-4 satisfies both conditions in Bunyakovsky conjecture if n == 1 mod 2)
{z}x: any length >= 2 (b^n-3 always satisfies both conditions in Bunyakovsky conjecture)
{z}y: any length >= 2 (b^n-2 always satisfies both conditions in Bunyakovsky conjecture)

These are the cyclotomic polynomials Phi(n,b) written in base b for n = 1 to 10000, since all cyclotomic polynomials satisfy both conditions in Bunyakovsky conjecture, it is conjectured that there are infinitely many bases b such that the number is prime, the smallest such base b are listed in [URL="https://oeis.org/A085398"]A085398[/URL] (note: 1 cannot be the base of a numeral system)

Interesting things: except GFN and GRU (see [URL="https://mersenneforum.org/showpost.php?p=568817&postcount=116"]this post[/URL]) (the smallest GFN in base b and the smallest GRU in base b is always both minimal prime (start with b+1) in base b and [URL="https://primes.utm.edu/glossary/xpage/UniquePrime.html"]unique prime[/URL] in base b (with period length "power of 2" and "prime", respectively), for some bases b, there are other numbers which are both minimal prime (start with b+1) in base b and unique prime in base b (the period length is neither prime nor power of 2)

* z1 (period = 6): see next post (if z1 is prime, then it is always such number, it is interesting not only because unique prime, but also because the Williams families z{0}1 and {z}1, whose 2-digit numbers are both z1)
* z0z1 (period = 10): 12, 20, 37, 38, 47, 71, 126, 131, 157, 158, 180, 223, 251, 257, 268, 276, 342, 361, 363, 375, 377, 385, 388, 438, 452, 462, 487, 507, 551, 588, 603, 628, 641, 652, 658, 676, 693, 707, 716, 738, 758, 760, 768, 782, 791, 792, 812, 850, 935, 973, 978, ... (z0z1 is prime, but z1, z01, zz1 are all composite)
* zz01 (period = 12): 12, 17, 27, 30, 38, 56, 61, 65, 94, 105, 109, 114, 126, 131, 166, 172, 183, 185, 196, 198, 225, 229, 231, 257, 261, 263, 269, 270, 296, 302, 308, 313, 324, 328, 339, 346, 350, 363, 374, 407, 412, 413, 424, 426, 429, 434, 437, 438, 443, 447, 463, 468, 480, 485, 502, 503, 507, 511, 515, 533, 545, 549, 558, 559, 571, 582, 593, 595, 599, 612, 614, 632, 638, 640, 641, 653, 659, 676, 699, 701, 707, 716, 718, 724, 725, 738, 742, 749, 767, 768, 777, 794, 797, 798, 801, 805, 811, 832, 842, 862, 868, 871, 872, 893, 905, 909, 923, 949, 953, 958, 963, 967, 974, 978, 1009, 1015, 1018 (zz01 is prime, but z1, z01, zz1 are all composite)

(see also [URL="https://mersenneforum.org/showpost.php?p=582061&postcount=154"]this post[/URL] for references of unique primes)

[CODE]
b primes which are both minimal prime (start with b+1) in base b and unique prime in base b
2 11 (3, period 2, both GFN and GRU)
3 12 (5, period 4, GFN), 21 (7, period 6), 111 (13, period 3, GRU)
4 11 (5, period 2, both GFN and GRU), 13 (7, period 3), 31 (13, period 6), 221 (41, period 10, this prime is completely a coincidence, since 41 = Phi(10,4)/gcd(Phi(10,4),10) and gcd(Phi(10,4),10) > 1)
5 12 (7, period 6), 23 (13, period 4, GFN), 111 (31, period 3, GRU)
6 11 (7, period 2, both GFN and GRU), 51 (31, period 6)
7 25 (19, period 3), 61 (43, period 6), 3334 (1201, period 8, GFN), 11111 (2801, period 5, GRU)
8 23 (19, period 6), 111 (73, period 3, GRU)
9 45 (41, period 4, GFN), 81 (73, period 6)
10 11 (11, period 2, both GFN and GRU), 37 (period 3)
11 34 (37, period 6), 61 (period 4, GFN)
12 11 (13, period 2, both GFN and GRU), B0B1 (19141, period 10), BB01 (20593, period 12)
[/CODE]

They are usually 2-digit numbers (with period 2, 3, 4, 6) (3 only for bases == 1 mod 3, and 4 only for bases == 1 mod 2), very few > 2-digit numbers.

sweety439 2022-05-12 12:59

The minimal prime in family 1{z} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family 1{0}z in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both 1z in base b

Such bases b are 2, 3, 4, 6, 7, 9, 10, 12, 15, 16, 19, 21, 22, 24, 27, 30, 31, 34, 36, 37, 40, 42, 45, 49, 51, 52, 54, 55, 57, 64, 66, 69, 70, 75, 76, 79, 82, 84, 87, 90, 91, 96, 97, 99, 100, 106, 112, 114, 115, 117, 120, 121, 126, 129, 132, 135, 136, 139, 141, 142, 147, 154, 156, 157, 159, 166, 169, 174, 175, 177, 180, 184, 187, 190, 192, 195, 199, 201, 205, 210, 211, 216, 217, 220, 222, 225, 229, 231, 232, 234, 240, 244, 246, 250, 252, 255, 261, 262, 271, 274, 279, 282, 285, 286, 289, 294, 297, 300, 301, 304, 307, 309, 310, 316, 321, 322, 324, 327, 330, 331, 337, 339, 342, 346, 351, 355, 360, 364, 367, 370, 372, 376, 379, 381, 385, 387, 394, 399, 405, 406, 411, 412, 414, 415, 420, 427, 429, 430, 432, 439, 441, 442, 444, 454, 456, 460, 465, 469, 471, 474, 477, 484, 486, 489, 492, 496, 499, 505, 507, 510, 511, 516, 517, 520, 525, 526, 531, 532, 535, 544, 546, 547, 549, 552, 555, 559, 562, 565, 576, 577, 582, 586, 591, 594, 597, 601, 607, 609, 612, 615, 616, 619, 625, 630, 639, 640, 642, 645, 646, 649, 651, 652, 654, 660, 661, 664, 681, 684, 687, 691, 700, 705, 712, 714, 715, 717, 720, 724, 726, 727, 730, 736, 741, 742, 744, 745, 747, 750, 756, 762, 766, 772, 775, 777, 780, 784, 786, 790, 792, 799, 801, 804, 805, 807, 810, 811, 814, 819, 829, 832, 834, 835, 847, 849, 850, 855, 861, 862, 867, 871, 874, 877, 880, 889, 892, 894, 895, 901, 906, 912, 916, 924, 931, 934, 936, 937, 939, 940, 945, 951, 954, 957, 966, 967, 975, 976, 987, 990, 994, 997, 999, 1000, 1002, 1006, 1009, 1014, 1015, 1020, ..., they are listed in [URL="https://oeis.org/A006254"]https://oeis.org/A006254[/URL], in these bases b, 1z is minimal prime (start with b+1)

The minimal prime in family {z}1 in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family z{0}1 in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both z1 in base b

Such bases b are 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 21, 22, 25, 28, 34, 39, 42, 51, 55, 58, 60, 63, 67, 70, 72, 76, 78, 79, 81, 90, 91, 100, 102, 106, 111, 112, 118, 120, 132, 139, 142, 144, 148, 151, 154, 156, 162, 163, 165, 168, 169, 174, 177, 189, 190, 193, 195, 204, 207, 210, 216, 219, 232, 237, 246, 247, 267, 273, 279, 280, 288, 289, 291, 294, 310, 315, 330, 333, 337, 343, 345, 349, 352, 358, 370, 379, 382, 384, 393, 396, 399, 403, 405, 406, 415, 417, 427, 435, 436, 448, 454, 456, 457, 477, 490, 496, 501, 513, 519, 526, 531, 532, 534, 538, 541, 552, 555, 561, 567, 568, 573, 580, 583, 585, 604, 606, 610, 613, 622, 625, 627, 636, 643, 645, 669, 672, 678, 687, 697, 702, 721, 727, 729, 736, 744, 748, 756, 762, 763, 769, 774, 783, 786, 793, 799, 802, 813, 819, 820, 826, 828, 837, 840, 847, 856, 858, 861, 865, 876, 879, 891, 895, 898, 900, 912, 916, 919, 921, 928, 951, 960, 961, 970, 975, 982, 988, 991, 993, 994, 1002, 1003, 1008, 1012, 1017, 1021, 1023, ..., they are listed in [URL="https://oeis.org/A055494"]https://oeis.org/A055494[/URL], in these bases b, z1 is minimal prime (start with b+1)

The minimal prime in family y{z} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family {y}z in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both yz in base b (note: yz does not exist in base 2, since it has leading zeros)

Such bases b are 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 20, 21, 22, 25, 27, 29, 31, 32, 36, 39, 40, 42, 45, 46, 47, 49, 51, 54, 55, 56, 57, 60, 61, 65, 66, 67, 69, 71, 77, 84, 86, 87, 90, 94, 95, 97, 101, 102, 104, 115, 116, 121, 126, 127, 131, 132, 135, 139, 141, 142, 145, 146, 149, 150, 154, 155, 156, 159, 160, 161, 164, 165, 170, 172, 175, 177, 181, 182, 185, 187, 189, 192, 194, 196, 197, 200, 204, 207, 210, 216, 219, 220, 221, 226, 231, 232, 234, 237, 241, 242, 245, 247, 249, 259, 260, 264, 265, 266, 269, 282, 286, 289, 291, 295, 297, 299, 302, 304, 306, 307, 310, 315, 320, 324, 331, 332, 336, 340, 344, 350, 351, 352, 355, 359, 361, 362, 365, 374, 375, 379, 381, 386, 387, 392, 394, 396, 397, 402, 406, 409, 419, 420, 421, 424, 429, 432, 434, 446, 449, 450, 451, 454, 456, 457, 462, 464, 469, 472, 474, 475, 476, 479, 482, 487, 491, 495, 496, 497, 500, 501, 502, 505, 507, 511, 512, 516, 519, 520, 529, 531, 542, 545, 549, 550, 551, 552, 561, 562, 570, 572, 574, 579, 582, 589, 590, 592, 595, 596, 597, 599, 605, 607, 611, 617, 619, 625, 626, 630, 634, 640, 641, 645, 647, 649, 652, 655, 656, 660, 669, 671, 674, 677, 684, 687, 692, 694, 696, 700, 706, 707, 709, 710, 716, 717, 721, 725, 729, 740, 744, 747, 751, 754, 755, 762, 764, 766, 770, 777, 780, 781, 782, 787, 795, 799, 804, 805, 810, 815, 816, 817, 826, 834, 839, 846, 849, 857, 864, 874, 882, 885, 886, 890, 891, 902, 907, 916, 922, 924, 926, 931, 940, 942, 945, 947, 951, 956, 957, 964, 966, 967, 969, 975, 977, 982, 985, 990, 995, 997, 999, 1001, 1002, 1006, 1007, 1014, 1015, 1017, 1024, ..., they are listed in [URL="https://oeis.org/A002328"]https://oeis.org/A002328[/URL], in these bases b, yz is minimal prime (start with b+1)

The minimal prime in family {z}y in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family z{y} in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both zy in base b (note: zy does not exist in base 2, since it has trailing zeros)

Such bases b are 3, 5, 7, 9, 13, 15, 19, 21, 27, 29, 33, 35, 37, 43, 47, 49, 55, 61, 63, 69, 71, 75, 77, 89, 93, 103, 107, 117, 119, 121, 127, 131, 135, 139, 145, 155, 161, 169, 173, 177, 183, 191, 205, 211, 217, 223, 231, 233, 237, 239, 247, 253, 257, 259, 265, 267, 273, 279, 285, 293, 299, 301, 303, 309, 313, 315, 323, 335, 337, 341, 355, 357, 359, 371, 387, 391, 399, 405, 415, 421, 425, 429, 435, 441, 443, 447, 449, 467, 469, 481, 489, 491, 495, 497, 505, 513, 523, 525, 531, 541, 545, 561, 565, 569, 573, 575, 583, 587, 595, 607, 609, 615, 621, 637, 645, 653, 663, 667, 671, 677, 713, 715, 719, 727, 733, 743, 747, 751, 757, 761, 763, 775, 779, 785, 789, 797, 807, 811, 817, 825, 841, 863, 875, 881, 897, 901, 931, 943, 945, 951, 959, 973, 979, 987, 993, 995, 1003, 1013, 1021, 1023, ..., they are listed in [URL="https://oeis.org/A028870"]https://oeis.org/A028870[/URL], in these bases b, zy is minimal prime (start with b+1)

The minimal prime in family {1} in base b (always minimal prime (start with b+1) base b) has length 2 if and only if the minimal prime in family 1{0}1 in base b (always minimal prime (start with b+1) base b) has length 2, since the corresponding primes are both 11 in base b

Such bases b are 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348, 352, 358, 366, 372, 378, 382, 388, 396, 400, 408, 418, 420, 430, 432, 438, 442, 448, 456, 460, 462, 466, 478, 486, 490, 498, 502, 508, 520, 522, 540, 546, 556, 562, 568, 570, 576, 586, 592, 598, 600, 606, 612, 616, 618, 630, 640, 642, 646, 652, 658, 660, 672, 676, 682, 690, 700, 708, 718, 726, 732, 738, 742, 750, 756, 760, 768, 772, 786, 796, 808, 810, 820, 822, 826, 828, 838, 852, 856, 858, 862, 876, 880, 882, 886, 906, 910, 918, 928, 936, 940, 946, 952, 966, 970, 976, 982, 990, 996, 1008, 1012, 1018, 1020, ..., they are listed in [URL="https://oeis.org/A006093"]https://oeis.org/A006093[/URL] (except 1, since 1 cannot be the base of a numeral system), in these bases b, 11 is minimal prime (start with b+1)

The minimal prime in family 1{0}11 in base b (always minimal prime (start with b+1) base b) has length 3 if and only if the minimal prime in family 11{0}1 in base b (always minimal prime (start with b+1) base b) has length 3, since the corresponding primes are both 111 in base b, and both of them do not have the length 2 number, 111 is minimal prime (start with b+1) in bases b = 3, 5, 8, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 69, 71, 75, 77, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 141, 143, 147, 153, 155, 161, 164, 167, 168, 173, 176, 188, 189, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278, 279, 287, 288, 290, 293, 309, 314, 329, 332, 342, 344, 351, 357, 369, 381, 383, 392, 395, 398, 402, 404, 405, 414, 416, 426, 434, 435, 447, 453, 455, 476, 489, 495, 500, 512, 518, 525, 530, 531, 533, 537, 551, 554, 560, 566, 567, 572, 579, 582, 584, 603, 605, 609, 621, 624, 626, 635, 644, 668, 671, 677, 686, 696, 701, 720, 728, 735, 743, 747, 755, 761, 762, 773, 782, 785, 792, 798, 801, 812, 818, 819, 825, 827, 836, 839, 846, 855, 857, 860, 864, 875, 878, 890, 894, 897, 899, 911, 915, 920, 927, 950, 959, 960, 969, 974, 981, 987, 992, 993, 1001, 1002, 1007, 1011, 1016, 1022, ..., they are the set difference [URL="https://oeis.org/A002384"]https://oeis.org/A002384[/URL] \ [URL="https://oeis.org/A006093"]https://oeis.org/A006093[/URL], and they are also the bases b such that the minimal prime in family {1} in base b (always minimal prime (start with b+1) base b) has length 3

The number "101" (in base b) is minimal prime (start with b+1) in these bases b:

14, 20, 24, 26, 54, 56, 74, 84, 90, 94, 110, 116, 120, 124, 134, 146, 160, 170, 176, 184, 204, 206, 224, 230, 236, 260, 264, 284, 300, 314, 326, 340, 350, 384, 386, 406, 436, 440, 444, 464, 470, 474, 496, 536, 544, 584, 594, 634, 636, 644, 654, 674, 680, 686, 696, 704, 714, 716, 740, 764, 780, 784, 816, 860, 864, 890, 920, 930, 950, 960, 986, 1004, 1010

They are the bases b such that ...

* The minimal prime in family 10{1} has length 3 (and "11" is not prime)
* The minimal prime in family 1{0}1 has length 3
* The minimal prime in family {1}01 has length 3 (and "11" is not prime)

The number "102" (in base b) is minimal prime (start with b+1) in these bases b:

33, 117, 123, 219, 243, 273, 297, 303, 321, 363, 369, 375, 423, 453, 513, 549, 573, 603, 609, 711, 753, 777, 867, 897, 903, 933, 957, ...

They are the bases b such that ...

* The minimal prime in family 10{2} has length 3 (and "12" is not prime)
* The minimal prime in family 1{0}2 has length 3
* The minimal prime in family {1}02 has length 3 (and "12" is not prime)

The number "201" (in base b) is minimal prime (start with b+1) in these bases b:

24, 27, 42, 45, 66, 72, 87, 93, 102, 123, 132, 162, 177, 201, 237, 240, 255, 264, 297, 324, 327, 351, 357, 360, 387, 399, 417, 423, 450, 456, 462, 474, 486, 489, 492, 537, 555, 570, 588, 597, 621, 633, 636, 657, 666, 693, 705, 750, 756, 759, 762, 768, 786, 792, 819, 837, 864, 885, 897, 918, 951, 960, 963, 978, 990, 1011, 1017, 1020, ...

They are the bases b such that ...

* The minimal prime in family 20{1} has length 3 (and "21" is not prime)
* The minimal prime in family 2{0}1 has length 3
* The minimal prime in family {2}01 has length 3 (and "21" is not prime)

The number "10z" (in base b) is minimal prime (start with b+1) in these bases b:

5, 8, 11, 13, 20, 26, 28, 35, 38, 39, 41, 44, 46, 48, 50, 53, 56, 59, 60, 65, 68, 83, 85, 86, 89, 93, 94, 101, 103, 125, 130, 131, 134, 138, 140, 144, 145, 148, 149, 153, 155, 158, 160, 163, 164, 171, 176, 181, 186, 188, 191, 193, 196, 203, 206, 209, 215, 218, 219, 230, 233, 236, 241, 248, 258, 259, 263, 264, 265, 268, 281, 288, 290, 296, 298, 303, 305, 306, 314, 319, 323, 335, 343, 349, 350, 354, 358, 361, 373, 374, 378, 380, 386, 391, 393, 395, 396, 401, 408, 418, 419, 423, 428, 431, 433, 445, 448, 449, 450, 453, 455, 461, 463, 468, 473, 475, 478, 481, 490, 494, 495, 500, 501, 504, 506, 515, 518, 519, 528, 530, 541, 548, 550, 551, 560, 561, 569, 571, 573, 578, 581, 588, 589, 595, 596, 598, 604, 606, 610, 618, 624, 629, 633, 644, 648, 655, 659, 668, 670, 673, 676, 683, 686, 693, 695, 699, 706, 708, 709, 716, 728, 739, 743, 746, 753, 754, 761, 763, 765, 769, 776, 779, 781, 794, 798, 803, 809, 815, 816, 825, 833, 838, 845, 848, 856, 863, 873, 881, 884, 885, 890, 915, 921, 923, 925, 930, 941, 944, 946, 950, 955, 956, 963, 965, 968, 974, 981, 984, 989, 996, 998, 1001, 1005, 1013, 1016, 1023, 1024, ...

They are the bases b such that ...

* The minimal prime in family 10{z} has length 3 (and "1z" is not prime)
* The minimal prime in family 1{0}z has length 3
* The minimal prime in family {1}0z has length 3 (and "1z" is not prime)

The number "z01" (in base b) is minimal prime (start with b+1) in these bases b:

5, 8, 14, 24, 26, 29, 33, 35, 36, 40, 43, 45, 48, 49, 50, 80, 83, 89, 93, 96, 99, 101, 104, 110, 115, 121, 124, 133, 135, 138, 140, 149, 161, 170, 173, 181, 191, 194, 201, 203, 205, 206, 209, 211, 215, 224, 226, 230, 253, 254, 258, 259, 260, 266, 274, 278, 281, 285, 286, 300, 301, 309, 311, 318, 320, 321, 323, 325, 334, 336, 344, 348, 353, 355, 365, 366, 373, 380, 386, 390, 395, 400, 404, 411, 414, 423, 425, 428, 433, 441, 444, 446, 451, 458, 465, 474, 483, 484, 489, 491, 495, 500, 505, 506, 514, 518, 523, 525, 528, 539, 560, 565, 566, 579, 584, 586, 594, 608, 616, 619, 623, 629, 654, 663, 666, 670, 679, 681, 695, 710, 714, 723, 734, 750, 754, 755, 764, 765, 773, 785, 789, 803, 806, 808, 815, 818, 824, 848, 853, 854, 855, 859, 873, 880, 881, 883, 884, 899, 908, 910, 925, 946, 954, 965, 969, 980, 983, 1011, 1014, ...

They are the bases b such that ...

* The minimal prime in family z0{1} has length 3 (and "z1" is not prime)
* The minimal prime in family z{0}1 has length 3
* The minimal prime in family {z}01 has length 3 (and "z1" is not prime)

The number "1zz" (in base b) is minimal prime (start with b+1) in these bases b:

8, 11, 13, 17, 18, 25, 28, 38, 39, 41, 43, 46, 50, 56, 59, 62, 63, 73, 80, 81, 85, 92, 95, 98, 102, 108, 109, 113, 118, 125, 127, 134, 137, 140, 143, 153, 155, 158, 160, 164, 165, 171, 172, 178, 179, 181, 183, 185, 186, 188, 196, 197, 200, 204, 206, 214, 228, 237, 238, 242, 245, 248, 249, 251, 256, 259, 263, 265, 266, 270, 273, 277, 280, 288, 291, 293, 295, 305, 311, 312, 323, 329, 332, 335, 353, 363, 365, 374, 375, 378, 384, 386, 391, 402, 409, 410, 416, 419, 426, 433, 435, 437, 440, 447, 448, 452, 455, 458, 459, 466, 468, 470, 472, 475, 482, 487, 498, 501, 508, 512, 514, 519, 528, 533, 539, 542, 556, 563, 567, 570, 571, 573, 580, 595, 602, 606, 608, 610, 617, 620, 622, 623, 634, 636, 650, 659, 671, 675, 679, 689, 690, 693, 696, 701, 703, 706, 708, 710, 713, 718, 721, 729, 731, 735, 739, 743, 749, 757, 759, 764, 774, 795, 797, 815, 816, 818, 820, 827, 839, 840, 848, 851, 854, 857, 869, 876, 882, 885, 893, 896, 899, 910, 918, 923, 930, 941, 944, 948, 958, 959, 962, 970, 973, 979, 981, 984, 993, 1001, 1004, 1016, 1018, ...

They are the bases b such that ...

* The minimal prime in family 1{z} has length 3
* The minimal prime in family {1}zz has length 3 (and "1z" is not prime)

The number "zz1" (in base b) is minimal prime (start with b+1) in these bases b:

10, 11, 14, 19, 26, 31, 35, 41, 45, 50, 52, 54, 57, 62, 75, 77, 84, 85, 87, 92, 95, 99, 101, 116, 125, 129, 130, 134, 136, 140, 141, 147, 155, 167, 176, 202, 221, 230, 239, 242, 244, 245, 252, 256, 259, 262, 264, 265, 271, 272, 274, 277, 286, 290, 292, 295, 297, 299, 305, 307, 316, 321, 322, 327, 329, 340, 347, 351, 354, 356, 362, 364, 372, 376, 391, 392, 414, 419, 441, 442, 446, 449, 455, 465, 466, 470, 472, 479, 482, 486, 491, 494, 497, 504, 510, 516, 521, 525, 540, 547, 554, 557, 570, 574, 584, 587, 591, 594, 605, 617, 619, 626, 629, 631, 642, 647, 650, 655, 656, 670, 682, 684, 685, 686, 689, 696, 712, 722, 732, 739, 741, 745, 746, 750, 752, 757, 759, 761, 766, 771, 776, 780, 787, 789, 790, 809, 815, 817, 824, 827, 831, 834, 836, 839, 851, 857, 860, 867, 869, 882, 901, 904, 915, 922, 924, 932, 934, 946, 950, 957, 959, 971, 972, 976, 977, 987, 1000, 1005, 1019, ...

They are the bases b such that ...

* The minimal prime in family {z}1 has length 3
* The minimal prime in family zz{1} has length 3 (and "z1" is not prime)

The number "yzz" (in base b) is minimal prime (start with b+1) in these bases b: (note: this family does not exist in base b=2)

13, 18, 30, 33, 48, 58, 75, 79, 85, 96, 99, 100, 106, 111, 114, 117, 118, 120, 129, 133, 138, 144, 153, 162, 171, 174, 186, 195, 199, 202, 222, 223, 243, 246, 252, 273, 276, 279, 298, 303, 328, 334, 342, 345, 348, 366, 372, 376, 378, 393, 400, 403, 418, 426, 435, 436, 447, 459, 460, 463, 465, 468, 471, 480, 493, 498, 504, 508, 510, 525, 526, 535, 541, 556, 558, 565, 567, 594, 612, 613, 636, 651, 667, 682, 688, 690, 702, 703, 705, 711, 732, 733, 736, 738, 745, 759, 763, 789, 790, 792, 796, 798, 831, 832, 835, 844, 855, 859, 862, 865, 871, 877, 898, 903, 915, 925, 933, 946, 948, 961, 970, 976, 981, 987, 993, 1000, 1009, ...

They are the bases b such that ...

* The minimal prime in family y{z} has length 3
* The minimal prime in family {y}zz has length 3 (and "yz" is not prime)

The number "zzy" (in base b) is minimal prime (start with b+1) in these bases b: (note: this family does not exist in base b=2)

31, 67, 91, 99, 109, 129, 151, 165, 187, 189, 201, 207, 241, 277, 289, 319, 367, 369, 411, 417, 427, 439, 445, 457, 459, 477, 511, 549, 555, 619, 651, 657, 669, 691, 697, 741, 745, 771, 787, 819, 859, 861, 879, 927, 939, 949, 967, 1015, ...

They are the bases b such that ...

* The minimal prime in family {z}y has length 3
* The minimal prime in family zz{y} has length 3 (and "zy" is not prime)

sweety439 2022-05-27 07:59

1 Attachment(s)
Upload newest pdf file.

sweety439 2022-05-27 09:18

[URL="http://factordb.com/index.php?id=1100000003573679860"]5(7^62668) (base b=11)[/URL] is PRP (strong PRP to bases 2, 3, 5, 7, 11), and (assuming its primality) it is a minimal prime (start with b+1) in base b = 11

sweety439 2022-05-28 06:11

This problem (the minimal prime (start with b+1) problem) is better than the CRUS problem because:

* [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] exclude some k's having primes (e.g. R14 k=1, 1*14^1-1 is prime, and R20 k=1, 1*20^1-1 is prime), even though they in fact have primes (the reason is that they only have this prime and have no other primes), as well as the extended [URL="https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub"]Sierpinski[/URL] and [URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]Riesel[/URL] problems, S8 k=27 has a prime (27*8^1+1)/gcd(27+1,8-1), and S16 k=4 has a prime (4*16^1+1)/gcd(4+1,16-1), and R4 k=1, R8 k=1, R16 k=1, R36 k=1, etc. all of them have primes, but still excluded in the problems (the reason is that they only have this prime and have no other primes)), in contrast, although the prime "111" is the only prime in the family {1} in base 8, it is still included in the minimal prime (start with b+1) problem in base b=8, since the prime "111" in base 8 is not only the prime for the family {1} in base 8 but also the prime for the family 1{0}11 and 11{0}1 in base 8, the same holds for the prime "11" in base 4, the prime "11" in base 4 is not only the prime for the family {1} in base 4 but also the prime for the family 1{0}1 in base 4, a more complex example is the prime "G7" in base 27 (= 439 in decimal), this prime is the only prime in the family {G}7 in base 27 (because of the difference-of-cubes algebraic factorization), but this prime is still included, since this prime is also the prime for the family G{7} in base 27, and the family G{7} in base 27 may contain infinitely many primes.

sweety439 2022-05-28 06:17

[URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]this table[/URL] includes the smallest prime (or PRP) in many classic families in various bases b:

* {1}: (b^n-1)/(b-1) (generalized repunit primes base b)
* 1{0}1: b^n+1 (generalized Fermat primes base b)
* {#}$: (b^n+1)/2 (generalized half Fermat primes base b)
* 1{0}2: b^n+2
* {z}y: b^n-2
* 1{0}3: b^n+3
* {z}x: b^n-3
* 1{0}4: b^n+4
* {z}w: b^n-4
* 2{0}1: 2*b^n+1
* 1{z}: 2*b^n-1
* 3{0}1: 3*b^n+1
* 2{z}: 3*b^n-1
* 4{0}1: 4*b^n+1
* 3{z}: 4*b^n-1
* 1{0}z: b^n+(b-1)
* {z}1: b^n-(b-1)
* z{0}1: (b-1)*b^n+1
* y{z}: (b-1)*b^n-1
* {y}z: ((b-2)*b^n+1)/(b-1)

also these families which may not be minimal primes (start with b+1) base b:

* 1{0}11: b^n+(b+1)
* {z}yz: b^n-(b+1)
* 11{0}1: (b+1)*b^n+1
* 10{z}: (b+1)*b^n-1

sweety439 2022-05-29 12:14

Records for the lengths of the numbers in these families in base b:

(the numbers in "()" are the lengths, not n)

{1}: ((b^n-1)/(b-1), length n, b >= 2, n >= 2)

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)
51 (4229)
91 (4421)
152 (270217)
185? (>66337)

1{0}1: (b^n+1, length n+1, b >= 2, n >= 1)

2 (2)
14 (3)
34 (5)
38 (>16777216)

1{0}2: (b^n+2, length n+1, b >= 3, n >= 1)

3 (2)
23 (12)
47 (114)
89 (256)
167 (>100001)

{z}y: (b^n-2, length n, b >= 3, n >= 2)

3 (2)
11 (4)
17 (6)
23 (24)
79 (38)
81 (130)
97 (747)
287 (3410)
305 (>30000)

1{0}3: (b^n+3, length n+1, b >= 4, n >= 1)

4 (2)
22 (3)
32 (4)
46 (21)
292 (40)
382 (256)
530 (1399)
646 (>5000)

{z}x: (b^n-3, length n, b >= 4, n >= 2)

4 (2)
16 (3)
22 (6)
28 (10)
50 (21)
52 (105)
94 (204)
152 (346)
154 (396)
302 (1061)
478 (1410)
512 (1600)
542 (1944)
1192 (>5000)

1{0}4: (b^n+4, length n+1, b >= 5, n >= 1)

5 (3)
23 (7)
53 (13403)
139? (>5000)

{z}w: (b^n-4, length n, b >= 5, n >= 2)

5 (5)
27 (7)
35 (13)
47 (65)
65 (175)
123 (299)
141 (395)
207 (>5000)

1{0}z: (b^n+(b-1), length n+1, b >= 2, n >= 1)

2 (2)
5 (3)
14 (17)
32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
173? (>10001)

{z}1: (b^n-(b-1), length n, b >= 2, n >= 2)

2 (2)
5 (5)
8 (13)
20 (17)
29 (33)
37 (67)
71 (3019)
93 (>60000)

1{0}11: (b^n+(b+1), length n+1, b >= 2, n >= 2)

2 (3)
9 (4)
11 (5)
23 (10)
35 (16)
63 (74)
68 (596)
198 (5198)
213 (>10001)

{z}yz: (b^n-(b+1), length n, b >= 2, n >= 2)

2 (3)
13 (4)
19 (5)
33 (7)
37 (9)
43 (31)
52 (108)
99 (131)
190 (562)
213 (643)
215 (22342)
517? (>5000)

2{0}1: (2*b^n+1, length n+1)



1{z}: (2*b^n-1, length n+1)



3{0}1: (3*b^n+1, length n+1)



2{z}: (3*b^n-1, length n+1)



4{0}1: (4*b^n+1, length n+1)



3{z}: (4*b^n-1, length n+1)



z{0}1: ((b-1)*b^n+1, length n+1)



y{z}: ((b-1)*b^n-1, length n+1)



11{0}1: ((b+1)*b^n+1, length n+2)



10{z}: ((b+1)*b^n-1, length n+2)



{y}z: (((b-2)*b^n+1)/(b-1), length n)

sweety439 2022-06-01 17:15

5 Attachment(s)
base 14 misses one prime: 4(D^19698)

the base 11 prime 5(7^62668) is only PRP, not proven prime

sweety439 2022-06-02 13:33

1 Attachment(s)
newest data file attached.

sweety439 2022-06-03 12:01

2 Attachment(s)
Base 15 is solved

Base 16 has these unsolved families:

{3}AF
{4}DD
D{B} (smallest prime has length 32235, i.e. the number D(B^32234))
D0{B} (not minimal prime (start with b+1) if the number of B's is >= 32234)
D00{B} (not minimal prime (start with b+1) if the number of B's is >= 32234)

sweety439 2022-06-03 12:13

[QUOTE=sweety439;593203]Using "sub-family", some unsolved families are obvious through [URL="https://en.wikiversity.org/wiki/Quasi-minimal_prime"]data for minimal primes (start with b+1) base b up to certain limit[/URL]: ("proper sub-family" is defined like "proper subset", i.e. sub-families other than itself)

Example 1: 80555551 = 80(5^5)1 is minimal prime (start with b+1) base b for b=10, thus all proper sub-families of 80{5}1 are unsolved families when searched to 5 5's if they [I]possible[/I] contain primes

Example 2: 55555025 = (5^5)25 is minimal prime (start with b+1) base b for b=8, thus all proper sub-families of {5}025 are unsolved families when searched to 5 5's if they [I]possible[/I] contain primes

Example 3: 33333301 = (3^6)01 is minimal prime (start with b+1) base b for b=7, thus all proper sub-families of {3}01 are unsolved families when searched to 6 3's if they [I]possible[/I] contain primes

Example 4: 100000000000507 = 1(0^11)507 is minimal prime (start with b+1) base b for b=9, thus all proper sub-families of 1{0}507 are unsolved families when searched to 11 0's if they [I]possible[/I] contain primes

Example 5: BBBBBB99B = (B^6)99B is minimal prime (start with b+1) base b for b=12, thus all proper sub-families of {B}99B are unsolved families when searched to 6 B's if they [I]possible[/I] contain primes

Example 6: 500025 = 5(0^3)25 is minimal prime (start with b+1) base b for b=8, thus all proper sub-families of 5{0}25 are unsolved families when searched to 3 0's if they [I]possible[/I] contain primes

Example 7: 77774444441 = (7^4)(4^6)1 is minimal prime (start with b+1) base b for b=8, thus:

* all proper sub-families of 7777{4}1 are unsolved families when searched to 6 4's if they [I]possible[/I] contain primes
* all proper sub-families of {7}4444441 are unsolved families when searched to 4 7's if they [I]possible[/I] contain primes

Example 8: 88888888833335 = (8^9)(3^4)5 is minimal prime (start with b+1) base b for b=9, thus:

* all proper sub-families of 888888888{3}5 are unsolved families when searched to 4 3's if they [I]possible[/I] contain primes
* all proper sub-families of {8}33335 are unsolved families when searched to 9 8's if they [I]possible[/I] contain primes

(Note: families with either [URL="https://en.wikipedia.org/wiki/Leading_zero"]leading zeros[/URL] or [URL="https://en.wikipedia.org/wiki/Trailing_zero"]trailing zeros[/URL] (or both) can be excluded, as they never produce primes > base, see [URL="https://mersenneforum.org/showpost.php?p=531660&postcount=10"]this related post[/URL])[/QUOTE]

The base 16 minimal prime (start with b+1) 300(F^1960)AF is a classic example of this (all proper sub-families of 300{F}AF are unsolved families when searched to 1960 F's if they [I]possible[/I] contain primes, but in fact, all proper sub-families of 300{F}AF can be ruled out as only contain composites (only count the numbers > base))

* 300{F}A: all numbers are divisible by 2
* 300{F}F: all numbers are divisible by 3
* 30{F}AF: the formula is 49*16^n-81, can be factored as difference of squares
* 00{F}AF: has leading zeros
* 300{F}: all numbers are divisible by 3
* 30{F}A: all numbers are divisible by 2
* 00{F}A: has leading zeros
* 30{F}F: all numbers are divisible by 3
* 00{F}F: has leading zeros
* 3{F}AF: the formula is 4*16^n-81, can be factored as difference of squares
* 0{F}AF: has leading zeros
* 30{F}: all numbers are divisible by 3
* 00{F}: has leading zeros
* 3{F}A: all numbers are divisible by 2
* 0{F}A: has leading zeros
* 3{F}F: all numbers are divisible by 3
* 0{F}F: has leading zeros
* {F}AF: all numbers are divisible by 5
* 3{F}: all numbers are divisible by 3
* 0{F}: has leading zeros
* {F}A: all numbers are divisible by 10
* {F}F: all numbers are divisible by 15
* {F}: all numbers are divisible by 15

sweety439 2022-06-03 12:19

1 Attachment(s)
Newest condensed table for bases 2 <= b <= 16 and b = 18 and b = 20:

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×b^n+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
9 151 3(0^1158)11 1161 3×9^1160+10
10 77 5(0^28)27 31 5×10^30+27
11 1068 5(7^62668) 62669 (57×11^62668−7)/10
12 106 4(0^39)77 42 4×12^41+91
13 3194~3197 8(0^32017)111 32021 8×13^32020+183
14 650 4(D^19698) 19699 5×14^19698−1
15 1284 (7^155)97 157 (15^157+59)/2
16 2345~2347 D(B^32234) 32235 (206×16^32234−11)/15
18 549 C(0^6268)5C 6271 12*18^6270+221
20 3314 G(0^6269)D 6271 16*20^6270+13
[/CODE]

Unsolved families:

base 13:

9{5}
A{3}A
C{5}C

base 16:

{3}AF
{4}DD

sweety439 2022-06-03 12:33

the "relative hardness" for base b is [URL="https://oeis.org/A062955"]A062955[/URL](b), i.e. (b-1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b), since [URL="https://oeis.org/A062955"]A062955[/URL](b) is the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([URL="https://en.wikipedia.org/wiki/Ordered_pair"]ordered pair[/URL]) of a minimal prime (start with b+1)) in base b (these (first digit,last digit) combos are also all possible (first digit,last digit) combos (ordered pair) of a prime > b in base b), since the first digit has b-1 choices (all digits except 0 can be the first digit), and the last digit has [URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b) choices (only digits [URL="https://en.wikipedia.org/wiki/Coprime"]coprime[/URL] to b (i.e. the digits in the [URL="https://en.wikipedia.org/wiki/Reduced_residue_system"]reduced residue system[/URL] mod b) can be the last digit), by the [URL="https://en.wikipedia.org/wiki/Rule_of_product"]rule of product[/URL], there are (b−1)*[URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](b) choices of the (first digit,last digit) combo, and both "number of minimal primes (start with b+1) base b" and "length of largest known minimal prime (start with b+1) base b" are [URL="https://en.wikipedia.org/wiki/Asymptotic_analysis"]roughly[/URL] [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://en.wikipedia.org/wiki/Euler%27s_constant"]gamma[/URL]*[URL="https://oeis.org/A062955"]A062955[/URL](b)), but for bases b = 7 and 15, the estimation of "length of largest known minimal prime (start with b+1) base b" is much higher than the real value, since these two bases are very-high [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] bases, these two bases are "primeful" as the [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] Sierpinski/Riesel conjectures in bases b = 7 and 15, while for bases b = 5, 11 and 14, the estimation of "length of largest known minimal prime (start with b+1) base b" is lower than the real value, since they are low [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] bases (as in [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL], bases == 2 mod 3 are low [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] bases), but the estimation of "number of minimal primes (start with b+1) base b" is always near to the real value

It can also be noted:

* In bases 3, 7, 15 (bases of the form 2^n-1), the largest minimal prime (start with b+1) starts with many digits (b-1)/2 (i.e. [B]111[/B] in base 3, [B]3333333333333333[/B]1 in base 7, [B]77777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777[/B]97 in base 15, I conjectured that this is true for all bases b of the form 2^n-1, since there are also many bases b of the form 2^n-1 (including 31, 63, 127, 255) without known [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]generalized half Fermat primes[/URL], and the smallest such primes must be minimal primes (start with b+1) in base b, and since the generalized half Fermat primes have very low weight and very low property to be primes (since the number of digits must be power of 2), they have a high property to be the largest minimal prime (start with b+1) to base b (assuming they exist), and they start with many digits (b-1)/2, followed by one digit (b+1)/2
* In bases 3, 7, 15 (bases of the form 2^n-1), the largest minimal prime (start with b+1) and the second-largest minimal prime (start with b+1) end with the same digit (1 for base 3, 1 for base 7, 7 for base 15), also in the case for base 15, both primes end with 97
* In bases 3, 7, 15 (bases of the form 2^n-1), the seconod-largest minimal prime (start with b+1) are special forms (dual Riesel (b^n-k, i.e. start with many digits b-1) for base 3 and base 15, Proth (k*b^n+1, i.e. end with many digits 0 followed by one digit 1) for base 7)

sweety439 2022-06-08 15:22

1 Attachment(s)
The product of all known minimal primes (start with b+1) base b: (I have added the "[URL="https://en.wikipedia.org/wiki/Aliquot_sequence"]Aliquot sequence[/URL]" "[URL="https://en.wikipedia.org/wiki/Home_prime"]Home prime[/URL] base b (only for the same base b)" "Inverse home prime base b (only for the same base b)" sequences in factordb starting with these numbers, they will be interesting!!! Also, I have already added "all these primes (or PRPs)" "all these primes (or PRPs)-1" "all these primes (or PRPs)+1" to factordb, as they are used as [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality proving[/URL] [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality proving[/URL] [URL="https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm"]P-1 integer factorization algorithm[/URL] [URL="https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm"]P+1 integer factorization algorithm[/URL])

(b = 13, 16, 17 have unsolved families)

(b = 11, 13, 16, 17, 22, 30 have unproven PRPs)

[URL="http://factordb.com/index.php?id=3"]b=2[/URL]

[URL="http://factordb.com/index.php?id=455"]b=3[/URL]

[URL="http://factordb.com/index.php?id=205205"]b=4[/URL]

[URL="http://factordb.com/index.php?id=1100000002457822814"]b=5[/URL]

[URL="http://factordb.com/index.php?id=1100000002457821560"]b=6[/URL]

[URL="http://factordb.com/index.php?id=1100000002457825324"]b=7[/URL]

[URL="http://factordb.com/index.php?id=1100000002371473795"]b=8[/URL]

[URL="http://factordb.com/index.php?id=1100000003450366253"]b=9[/URL]

[URL="http://factordb.com/index.php?id=1100000002370859491"]b=10[/URL]

[URL="http://factordb.com/index.php?id=1100000003583737715"]b=11[/URL]

[URL="http://factordb.com/index.php?id=1100000002457818232"]b=12[/URL]

[URL="http://factordb.com/index.php?id=1100000003782980695"]b=13[/URL]

[URL="http://factordb.com/index.php?id=1100000003575953976"]b=14[/URL]

[URL="http://factordb.com/index.php?id=1100000003588261354"]b=15[/URL]

[URL="http://factordb.com/index.php?id=1100000003793382618"]b=16[/URL]

[URL="http://factordb.com/index.php?id=1100000003798135482"]b=17[/URL]

[URL="http://factordb.com/index.php?id=1100000003590551018"]b=18[/URL]

[URL="http://factordb.com/index.php?id=1100000003590550972"]b=20[/URL]

[URL="http://factordb.com/index.php?id=1100000003798118963"]b=22[/URL]

[URL="http://factordb.com/index.php?id=1100000003782956267"]b=24[/URL]

[URL="http://factordb.com/index.php?id=1100000003782953720"]b=30[/URL]

sweety439 2022-06-09 14:18

New link for the minimal set of the primes > b in base b: [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]https://github.com/xayahrainie4793/quasi-mepn-data[/URL]

This site includes bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, all are completely solved (if we allow probable primes in place of proven primes) except base 13 and base 16, and both of these two bases have 2 unsolved families, bases 11, 18, 20, 22, 24, 30 needs primality proving of the probable primes (base 18 and base 20 are easier, since their largest probable primes only have 7872 and 8159 decimal digits, respectively)

The minimal set of the primes > b in base b must includes: (if such prime exists in base b)

* The smallest prime of the form (b^n-1)/(b-1) with n>=2 (i.e. the smallest generalized repunit prime in base b), see [URL="https://oeis.org/A084740"]https://oeis.org/A084740[/URL], [URL="https://oeis.org/A084738"]https://oeis.org/A084738[/URL], [URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL], [URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL], [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL], [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html[/URL], [URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL], [URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL]
* The smallest prime of the form b^n+1 with n>=1 (i.e. the smallest generalized Fermat prime in base b), see [URL="https://oeis.org/A228101"]https://oeis.org/A228101[/URL], [URL="https://oeis.org/A079706"]https://oeis.org/A079706[/URL], [URL="https://oeis.org/A084712"]https://oeis.org/A084712[/URL], [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL], [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL], [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL]
* The smallest prime of the form 2*b^n+1 with n>=1 (see [URL="https://oeis.org/A119624"]https://oeis.org/A119624[/URL], [URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL], [URL="https://mersenneforum.org/showthread.php?t=6918"]https://mersenneforum.org/showthread.php?t=6918[/URL], [URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL])
* The smallest prime of the form 2*b^n-1 with n>=1 (see [URL="https://oeis.org/A119591"]https://oeis.org/A119591[/URL], [URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL], [URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL], [URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL])
* The smallest prime of the form b^n+2 with n>=1 (see [URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL], [URL="https://oeis.org/A084713"]https://oeis.org/A084713[/URL])
* The smallest prime of the form b^n-2 with n>=2 (see [URL="https://oeis.org/A250200"]https://oeis.org/A250200[/URL], [URL="https://oeis.org/A255707"]https://oeis.org/A255707[/URL], [URL="https://oeis.org/A084714"]https://oeis.org/A084714[/URL], [URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL])
* The smallest prime of the form (b-1)*b^n+1 with n>=1 (see [URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL], [URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL], [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL], [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL], [URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL], [URL="http://www.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL])
* The smallest prime of the form (b-1)*b^n-1 with n>=1 (see [URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL], [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL], [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL], [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]https://www.rieselprime.de/ziki/Williams_prime_MM_table[/URL], [URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL], [URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL], [URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL], [URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL])
* The smallest prime of the form b^n+(b-1) with n>=1 (see [URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL], [URL="https://oeis.org/A076846"]https://oeis.org/A076846[/URL], [URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL])
* The smallest prime of the form b^n-(b-1) with n>=2 (see [URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL], [URL="https://oeis.org/A343589"]https://oeis.org/A343589[/URL], [URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL], [URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL], [URL="http://www.bitman.name/math/table/435"]http://www.bitman.name/math/table/435[/URL])

xilman 2022-06-11 13:26

Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums$
[/code]

Should be easy enough to fix but not a good advertisement ...

xilman 2022-06-11 14:34

[QUOTE=xilman;607581]Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums$
[/code]

Should be easy enough to fix but not a good advertisement ...[/QUOTE]Making it compile was easy but tedious.

However, the code is riddled with dubious string assignments which elicit numerous warnings from gcc. When the resulting binary runs the output is corrrect but incomplete. For instance, the output for base 6 is 11, 15, 21, 25, 31, 35, 45, 51 and longer strings are omitted. For base 10, nothing longer than 991 appears.

Not impressed.

sweety439 2022-06-11 14:38

[QUOTE=xilman;607581]Did anyone try building sweety439's C code from the first page of this thread? I cut and pasted the text into minimumprimes.c and tried to compile it.

An attempt just now produced this result:
[code]
pcl@thoth:~/nums$ gcc -O -o minimumprimes minimumprimes.c
minimumprimes.c: In function ‘isprime’:
minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)
| ^
minimumprimes.c: In function ‘hasdivisor’:
minimumprimes.c:510:33: warning: comparison between pointer and integer
510 | if(temp > base)
| ^
minimumprimes.c:941:14: error: conflicting types for ‘residues’; have ‘char[42]’
941 | char residues[42] = {1};
| ^~~~~~~~
minimumprimes.c:897:14: note: previous definition of ‘residues’ with type ‘char[30]’
897 | char residues[30] = {1};
| ^~~~~~~~
minimumprimes.c:965:13: error: redefinition of ‘coprimeres’
965 | int coprimeres = 0;
| ^~~~~~~~~~
minimumprimes.c:921:13: note: previous definition of ‘coprimeres’ with type ‘int’
921 | int coprimeres = 0;
| ^~~~~~~~~~
pcl@thoth:~/nums$
[/code]

Should be easy enough to fix but not a good advertisement ...[/QUOTE]

Well, this code need to run with GMP.

also, I contacted others, and he made a C++ code (with GMP) to run, and he ran this code for bases 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, but for bases 17, 19, 21, ..., this code seems to be very slow, he has sent the data to me, see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]my GitHub page[/URL]

xilman 2022-06-11 14:44

[QUOTE=sweety439;607587]Well, this code need to run with GMP.

also, I contacted others, and he made a C++ code (with GMP) to run, and he ran this code for bases 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30, but for bases 17, 19, 21, ..., this code seems to be very slow, he has sent the data to me, see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]my GitHub page[/URL][/QUOTE]Of course it was linked with GMP!

I do know how to write and build GMP-enabled programs.

To be honest, given that your code doesn't even compile, I had no confidence that it would do anything useful after being hacked to fix the egregious errors listed in my earlier post.

Why on earth should anyone want to check some of your numbers for primality, given this dismal record?

sweety439 2022-06-11 14:49

[QUOTE=xilman;607588]Of course it was linked with GMP!

I do know how to write and build GMP-enabled programs.

To be honest, given that your code doesn't even compile, I had no confidence that it would do anything useful after being hacked to fix the egregious errors listed in my earlier post.

Why on earth should anyone want to check some of your numbers for primality, given this dismal record?[/QUOTE]

This code is not created by me, this is I copied others' code for the original minimal prime problem (i.e. prime > base is not needed) and try to update the code for my new problem (i.e. only consider the primes > base), but there may be errors.

paulunderwood 2022-06-11 14:55

[QUOTE=sweety439;607589]This code is not created by me, this is I copied others' code for the original minimal prime problem (i.e. prime > base is not needed) and try to update the code for my new problem (i.e. only consider the primes > base), but there may be errors.[/QUOTE]

Hmm.

[CODE]minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)[/CODE]

Looks like [C]temp[/C] is [C]mpz_t[/C] and neither [C]mpz_cmp[/C] nor [C]mpz_cmp_ui[/C] were used for [C]temp > base[/C].

xilman 2022-06-11 15:09

[QUOTE=paulunderwood;607590]Hmm.

[CODE]minimumprimes.c:295:53: warning: comparison between pointer and integer
295 | if(mpz_probab_prime_p(temp, 25) > 0 && temp > base)[/CODE]

Looks like [C]temp[/C] is [C]mpz_t[/C] and neither [C]mpz_cmp[/C] nor [C]mpz_cmp_ui[/C] were used for [C]temp > base[/C].[/QUOTE]Exactly. The first one I fixed.

My code reads

[code]
if(mpz_probab_prime_p(temp, 25) > 0 && mpz_cmp_si(temp, base) > 0) /* PCL */
[/code]

Note mpz_cmp_si --- base is declared int, not unsigned.

sweety439 2022-06-24 06:09

[QUOTE=sweety439;606692]This problem (the minimal prime (start with b+1) problem) is better than the CRUS problem because:

* [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] exclude some k's having primes (e.g. R14 k=1, 1*14^1-1 is prime, and R20 k=1, 1*20^1-1 is prime), even though they in fact have primes (the reason is that they only have this prime and have no other primes), as well as the extended [URL="https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub"]Sierpinski[/URL] and [URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]Riesel[/URL] problems, S8 k=27 has a prime (27*8^1+1)/gcd(27+1,8-1), and S16 k=4 has a prime (4*16^1+1)/gcd(4+1,16-1), and R4 k=1, R8 k=1, R16 k=1, R36 k=1, etc. all of them have primes, but still excluded in the problems (the reason is that they only have this prime and have no other primes)), in contrast, although the prime "111" is the only prime in the family {1} in base 8, it is still included in the minimal prime (start with b+1) problem in base b=8, since the prime "111" in base 8 is not only the prime for the family {1} in base 8 but also the prime for the family 1{0}11 and 11{0}1 in base 8, the same holds for the prime "11" in base 4, the prime "11" in base 4 is not only the prime for the family {1} in base 4 but also the prime for the family 1{0}1 in base 4, a more complex example is the prime "G7" in base 27 (= 439 in decimal), this prime is the only prime in the family {G}7 in base 27 (because of the difference-of-cubes algebraic factorization), but this prime is still included, since this prime is also the prime for the family G{7} in base 27, and the family G{7} in base 27 may contain infinitely many primes.[/QUOTE]

This problem (the minimal prime (start with b+1) problem) covers these problems:

* Find the smallest prime of the form (b^n-1)/(b-1) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form (b^n+1)/2 (for odd b) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 2*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 2*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+2 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-2 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 3*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 3*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+3 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-3 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form 4*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form 4*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+4 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-4 with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form (b-1)*b^n+1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form (b-1)*b^n-1 with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n+(b-1) with n>=1 (or prove that such primes do not exist)
* Find the smallest prime of the form b^n-(b-1) with n>=2 (or prove that such primes do not exist)
* Find the smallest prime of the form ((b-2)*b^n+1)/(b-1) with n>=2 (or prove that such primes do not exist)

sweety439 2022-06-24 06:19

Now these minimal primes (start with b+1) in base b have been proven primes: (only list the numbers > 10^1000)

[CODE]
b index of this minimal prime in base b base-b form of the minimal prime algebraic ((a*b^n+c)/d) form of the minimal prime primality certificate for the minimal prime
9 151 3(0^1158)11 3×9^1160+10 [URL="http://factordb.com/cert.php?id=1100000002376318423"]http://factordb.com/cert.php?id=1100000002376318423[/URL]
11 1067 55(7^1011) (607×11^1011−7)/10 [URL="http://factordb.com/cert.php?id=1100000002361376522"]http://factordb.com/cert.php?id=1100000002361376522[/URL]
13 3184 (9^968)B (3×13^969+5)/4 [URL="http://factordb.com/cert.php?id=1100000000258566244"]http://factordb.com/cert.php?id=1100000000258566244[/URL]
13 3185 1(0^1295)181 13^1298+274 [URL="http://factordb.com/cert.php?id=1100000002615445013"]http://factordb.com/cert.php?id=1100000002615445013[/URL]
13 3186 (9^1362)5 (3×13^1363−19)/4 [URL="http://factordb.com/cert.php?id=1100000002321017776"]http://factordb.com/cert.php?id=1100000002321017776[/URL]
13 3187 (7^1504)1 (7×13^1505−79)/12 [URL="http://factordb.com/cert.php?id=1100000002320890755"]http://factordb.com/cert.php?id=1100000002320890755[/URL]
13 3188 93(0^1551)1 120×13^1552+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
13 3189 72(0^2297)2 93×13^2298+2 [URL="http://factordb.com/cert.php?id=1100000002632396910"]http://factordb.com/cert.php?id=1100000002632396910[/URL]
13 3190 177(0^2703)17 267×13^2705+20 [URL="http://factordb.com/cert.php?id=1100000003590430825"]http://factordb.com/cert.php?id=1100000003590430825[/URL]
13 3191 39(0^6266)1 48×13^6267+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
13 3192 B(0^6540)BBA 11×13^6543+2012 [URL="http://factordb.com/cert.php?id=1100000002616382906"]http://factordb.com/cert.php?id=1100000002616382906[/URL]
13 3193 (C^10631)92 13^10633−50 [URL="http://factordb.com/cert.php?id=1100000003590493750"]http://factordb.com/cert.php?id=1100000003590493750[/URL]
14 650 4(D^19698) 5×14^19698−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
16 2337 D(9^1052) (68×16^1052−3)/5 [URL="http://factordb.com/cert.php?id=1100000002321036020"]http://factordb.com/cert.php?id=1100000002321036020[/URL]
16 2338 FA(F^1062)45 251×16^1064−187 [URL="http://factordb.com/cert.php?id=1100000003588387610"]http://factordb.com/cert.php?id=1100000003588387610[/URL]
16 2339 F(8^1517)F (233×16^1518+97)/15 [URL="http://factordb.com/cert.php?id=1100000000633744824"]http://factordb.com/cert.php?id=1100000000633744824[/URL]
16 2340 2(0^1713)321 2×16^1716+801 [URL="http://factordb.com/cert.php?id=1100000003588386735"]http://factordb.com/cert.php?id=1100000003588386735[/URL]
16 2341 300(F^1960)AF 769×16^1962−81 [URL="http://factordb.com/cert.php?id=1100000003588368750"]http://factordb.com/cert.php?id=1100000003588368750[/URL]
16 2342 9(0^3542)91 9×16^3544+145 [URL="http://factordb.com/cert.php?id=1100000000633424191"]http://factordb.com/cert.php?id=1100000000633424191[/URL]
16 2343 5B(C^3700)D (459×16^3701+1)/5 [URL="http://factordb.com/cert.php?id=1100000000993764322"]http://factordb.com/cert.php?id=1100000000993764322[/URL]
18 549 C(0^6268)C5 12×18^6270+221 [URL="http://factordb.com/cert.php?id=1100000003590442437"]http://factordb.com/cert.php?id=1100000003590442437[/URL]
20 3312 5(0^1163)AJ 5×20^1165+219 [URL="http://factordb.com/cert.php?id=1100000003590502412"]http://factordb.com/cert.php?id=1100000003590502412[/URL]
20 3313 C(D^2449) (241×20^2449−13)/19 [URL="http://factordb.com/cert.php?id=1100000002325393915"]http://factordb.com/cert.php?id=1100000002325393915[/URL]
20 3314 G(0^6269)D 16×20^6270+13 [URL="http://factordb.com/cert.php?id=1100000003590539457"]http://factordb.com/cert.php?id=1100000003590539457[/URL]
22 7998 K(0^760)EC1 20×22^763+7041 [URL="http://factordb.com/cert.php?id=1100000000632724415"]http://factordb.com/cert.php?id=1100000000632724415[/URL]
22 7999 J(0^767)IGGJ 19×22^771+199779 [URL="http://factordb.com/cert.php?id=1100000003591362567"]http://factordb.com/cert.php?id=1100000003591362567[/URL]
22 8000 (7^959)K7 (22^961+857)/3 [URL="http://factordb.com/cert.php?id=1100000003591361817"]http://factordb.com/cert.php?id=1100000003591361817[/URL]
22 8001 (L^2385)KE7 22^2388−653 [URL="http://factordb.com/cert.php?id=1100000003591360774"]http://factordb.com/cert.php?id=1100000003591360774[/URL]
22 8002 (7^3815)2L (22^3817−289)/3 [URL="http://factordb.com/cert.php?id=1100000003591359839"]http://factordb.com/cert.php?id=1100000003591359839[/URL]
24 3405 (N^2644)LLN 24^2647−1201 [URL="http://factordb.com/cert.php?id=1100000003593270089"]http://factordb.com/cert.php?id=1100000003593270089[/URL]
24 3406 (D^2698)LD (13×24^2700+4403)/23 [URL="http://factordb.com/cert.php?id=1100000003593269876"]http://factordb.com/cert.php?id=1100000003593269876[/URL]
24 3407 A(0^2951)8ID 10×24^2954+5053 [URL="http://factordb.com/cert.php?id=1100000003593269654"]http://factordb.com/cert.php?id=1100000003593269654[/URL]
24 3408 88(N^5951) 201×24^5951−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
24 3409 N00(N^8129)LN 13249×24^8131−49 [URL="http://factordb.com/cert.php?id=1100000003593391606"]http://factordb.com/cert.php?id=1100000003593391606[/URL]
30 2616 C(0^1022)1 12×30^1023+1 proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N−1 test[/URL], since N−1 is trivially 100% factored
30 2617 (5^4882)J (5×30^4883+401)/29 [URL="http://factordb.com/cert.php?id=1100000002327649423"]http://factordb.com/cert.php?id=1100000002327649423[/URL]
30 2619 O(T^34205) 25×30^34205−1 proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], since N+1 is trivially 100% factored
[/CODE]

and I also used the Windows version of Primo to prove the smaller minimal primes (start with b+1), see [URL="http://factordb.com/certoverview.php?userid=1266"]http://factordb.com/certoverview.php?userid=1266[/URL]

These numbers are only probable primes: (they are all unproven probable primes for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30) (all of them are [URL="https://primes.utm.edu/glossary/xpage/StrongPRP.html"]strong probable primes[/URL] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, and [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]trial factored[/URL] to 10^11

[CODE]
b index of this minimal prime in base b (assuming the primality of all probable primes in base b) base-b form of the unproven probable prime algebraic ((a*b^n+c)/d) form of the unproven probable prime
11 1068 5(7^62668) (57×11^62668−7)/10
13 3194 C(5^23755)C (149×13^23756+79)/12
13 3195 8(0^32017)111 8×13^32020+183
16 2344 D0(B^17804) (3131×16^17804−11)/15
16 2345 D(B^32234) (206×16^32234−11)/15
22 8003 B(K^22001)5 (251×22^22002−335)/21
30 2618 I(0^24608)D 18×30^24609+13
[/CODE]

sweety439 2022-07-01 19:13

For a sub-problem of this problem, finding the minimal (a,b,c) triple (i.e. there is no a' <= a, b' <= b, c' <= c, except the case a' = a and b' = b and c' = c) such that xxx...xxxyyy...yyyzzz...zzz (with a x's, b y's, c z's) is prime

e.g. in base 8, the family (7^a)(4^b)1

the minimal pairs of (a,b) are:

(0,8)
(1,7)
(4,6)
(12,0)

they corresponding to minimal primes (start with base+1) 444444441, 744444441, 77774444441, 7777777777771, respectively.

(for a = 2, the minimal b is infinity (since all numbers 77(4^b)1 are divisible by 5); for a = 3, the minimal b is 11; for a = 5, the minimal b is 143; for a = 6, the minimal b is infinity (since all numbers 777777(4^b)1 are divisible by 5); for a = 7, the minimal b is 17; for a = 8, the minimal b is 16; for a = 9, the minimal b is 15; for a = 10, the minimal b is infinity (since all numbers 7777777777(4^b)1 are divisible by 5); for a = 11, the minimal b is 97; for b = 1, the minimal a is 79; for b = 2, the minimal a is 84; for b = 3, the minimal a is 233; for b = 4, the minimal a is 56; for b = 5, the minimal a is infinity (since all numbers (7^a)444441 are divisible by 7); thus none of them can produce minimal primes (start with base+1))

e.g. in base 9, the family (8^a)(3^b)5

the minimal pairs of (a,b) are:

(1,8)
(9,4)
(19,2)

they corresponding to minimal primes (start with base+1) 8333333335, 88888888833335, 8888888888888888888335, respectively.

sweety439 2022-07-01 19:24

Some minimal primes (start with b+1) in power-of-2 bases related to Sierpinski problem, Riesel problem, dual Sierpinski problem, dual Riesel problem (see [URL="https://oeis.org/A046067"]https://oeis.org/A046067[/URL], [URL="https://oeis.org/A046069"]https://oeis.org/A046069[/URL], [URL="https://oeis.org/A067760"]https://oeis.org/A067760[/URL], [URL="https://oeis.org/A096502"]https://oeis.org/A096502[/URL], also see [URL="http://www.prothsearch.com/"]http://www.prothsearch.com/[/URL]):

Base 8:

47777: a Riesel prime for k=5 (5*2^n-1)
7777777777771: smallest dual Riesel prime for k=7 (2^n-7)
777777777777777777777777777777777777777777777777777777777777777777777777777777777777441 (although not minimal prime (start with b+1) in base b = 8, but still appear in the proof of minimal prime (start with b+1) problem in base b = 8): smallest dual Riesel prime for k=223 (2^n-223)

Base 16:

40000000000000000000085: a dual Proth prime for k=133 (2^n+133)
CAFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF: a Riesel prime for k=203 (203*2^k-1)
52000000000000000000000000000000000000000000000000000000000000000000000001: a Proth prime for k=41 (41*2^n+1)
3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF23: smallest dual Riesel prime for k=221 (2^n-221)
8888FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF: a Riesel prime for k=34953 (34953*2^n-1)
88FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF: a Riesel prime for k=137 (137*2^n-1)
2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000321: a dual Proth prime for k=801 (2^n+801)

sweety439 2022-07-02 20:53

We have completely solved the "minimal prime > base problem" in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24

Also, we have completely solved the "minimal prime > base problem" in the weaker case that "a number > 20000 decimal digits passes the strong primality test to all primes bases <= 61 (see [URL="https://oeis.org/A014233"]https://oeis.org/A014233[/URL] and [URL="https://primes.utm.edu/glossary/xpage/StrongPRP.html"]https://primes.utm.edu/glossary/xpage/StrongPRP.html[/URL]) and passes the strong Lucas primality test with parameters (P, Q) defined by Selfridge's Method A (see [URL="https://oeis.org/A217255"]https://oeis.org/A217255[/URL] and [URL="http://ntheory.org/pseudoprimes.html"]http://ntheory.org/pseudoprimes.html[/URL]) and trial factored to 10^11 (see [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]https://primes.utm.edu/glossary/xpage/TrialDivision.html[/URL])" can be regarded as primes, in bases 11, 22, 30, these unproven probable primes for bases 11, 22, 30 are:

[CODE]
b base-b form of PRP algebraic form of PRP
11 5(7^62668) (57×11^62668−7)/10
22 B(K^22001)5 (251×22^22002−335)/21
30 I(0^24608)D 18×30^24609+13
[/CODE]

("these three numbers are in fact composite" will only cause the unsolved families 5{7} in base 11, B{K}5 in base 22, I{0}D in base 30, respectively)

(however, see [URL="https://primes.utm.edu/notes/prp_prob.html"]https://primes.utm.edu/notes/prp_prob.html[/URL], bases 11, 22, 30 are in fact 99.999999999999...% (with over 10000 9's) solved, although not 100% solved, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are 100% solved (thus, bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 have "minimal prime > base theorem"), thus for example, we cannot definitely say that base 11 has 1068 minimal primes (start with b+1) (although this is very likely, and we can definitely say that base 11 has either 1067 or 1068 minimal primes (start with b+1), and base 11 has 1067 minimal primes (start with b+1) if and only if 5(7^62668) is in fact composite and there is no prime of the form 5{7} in base 11, but both are very impossible), and we cannot definitely say that the largest minimal primes (start with b+1) in base 11 has length 62669 (although this is very likely, and we can definitely say that the largest minimal primes (start with b+1) in base 11 has length either 1013 or >=62669, it is 1013 if and only if 5(7^62668) is in fact composite and there is no prime of the form 5{7} in base 11, and it is n (n>62669) if and only if 5(7^62668) is in fact composite and there is a larger prime of the form 5{7} in base 11 and this prime has length n), however, we [B]can[/B] definitely say that base 24 has 3409 minimal primes (start with b+1), and we [B]can[/B] definitely say that the largest minimal primes (start with b+1) in base 24 has length 8134, since all these primes are proven primes.

Besides, for bases 13 and 16, the "minimal prime > base problem" is completely solved with the exception of these 3 families of the form x{y}z (again, in the weaker case that "a number > 20000 decimal digits passes the strong primality test to all primes bases <= 61 (see [URL="https://oeis.org/A014233"]https://oeis.org/A014233[/URL] and [URL="https://primes.utm.edu/glossary/xpage/StrongPRP.html"]https://primes.utm.edu/glossary/xpage/StrongPRP.html[/URL]) and passes the strong Lucas primality test with parameters (P, Q) defined by Selfridge's Method A (see [URL="https://oeis.org/A217255"]https://oeis.org/A217255[/URL] and [URL="http://ntheory.org/pseudoprimes.html"]http://ntheory.org/pseudoprimes.html[/URL]) and trial factored to 10^11 (see [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]https://primes.utm.edu/glossary/xpage/TrialDivision.html[/URL])" can be regarded as primes):

[CODE]
b base-b form of unsolved family base b algebraic form of unsolved family base b
13 9{5} (113×13^n−5)/12
13 A{3}A (41×13^(n+1)+27)/4
16 {3}AF (16^(n+2)+619)/5
[/CODE]

(also, even when we find a prime in these three families, that number will be so big that proving its primality can cost weeks (or months or years ...), like the status of the large unproven probable primes in this project, since these three families are not of the form *{0}1 or *{b-1}, thus neither [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1[/URL] nor [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1[/URL] can be >= 25% factored, and instead [URL="https://primes.utm.edu/prove/prove4_2.html"]ECPP primality proving[/URL] such as [URL="http://www.ellipsa.eu/public/primo/primo.html"]PRIMO[/URL] must be used)

Besides, there are unproven probable primes for bases 13 and 16:

[CODE]
b base-b form of PRP algebraic form of PRP
13 C(5^23755)C (149×13^23756+79)/12
13 8(0^32017)111 8×13^32020+183
16 D(B^32234) (206×16^32234−11)/15
16 (4^72785)DD (4×16^72787+2291)/15
[/CODE]

The "number of minimal primes (start with b+1)" in base 13 is:

3197 (likely)
3195~3197 (if strong PRP can be regarded as primes, and this is 99.999999999999...% (with over 10000 9's))
3193~3197 (definitely say, since C(5^23755)C is in fact composite will only cause an unsolved family C{5}C, and 8(0^32017)111 is in fact composite will only cause an unsolved family 8{0}111, so there [B]definitely[/B] cannot be more than 3197 minimal primes (start with b+1) in base 13)

The "number of minimal primes (start with b+1): in base 16 is:

2347 (likely)
2346~2347 (if strong PRP can be regarded as primes, and this is 99.999999999999...% (with over 10000 9's))
2343~2347 (definitely say, since D(B^32234) is in fact composite will only cause an unsolved family D{B}, and (4^72785)DD is in fact composite will only cause an unsolved family {4}DD, so there [B]definitely[/B] cannot be more than 2347 minimal primes (start with b+1) in base 16)

Condensed table: (bases 11, 13, 16, 17, 19, 21, 22, 26, 28, 30, 36 data assume the primality of the strong probable primes)

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×b^n+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
9 151 3(0^1158)11 1161 3×9^1160+10
10 77 5(0^28)27 31 5×10^30+27
11 1068 5(7^62668) 62669 (57×11^62668−7)/10
12 106 4(0^39)77 42 4×12^41+91
13 3195~3197 8(0^32017)111 32021 8×13^32020+183
14 650 4(D^19698) 19699 5×14^19698−1
15 1284 (7^155)97 157 (15^157+59)/2
16 2347 (3^116137)AF 116139 (16^116137+619)/5
17 10405~10428 G(7^32072)F 32074 (263×17^32073+121)/16
18 549 C(0^6268)5C 6271 12×18^6270+221
19 31400~31435 D17D(0^19750)1 19755 89674×19^19751+1
20 3314 G(0^6269)D 6271 16×20^6270+13
21 13373~13395 5D(0^19848)1 19851 118×21^19849+1
22 8003 B(K^22001)5 22003 (251×22^22002−335)/21
24 3409 N00(N^8129)LN 8134 13249×24^8131−49
26 25250~25259 (5^19391)6F 19393 (26^19393+179)/5
28 25526~25529 N(6^24051)LR 24054 (209×28^24053+3967)/9
30 2619 O(T^34205) 34206 25×30^34205−1
36 35256~35263 (J^10117)LJ 10119 (19×36^10119+2501)/35
[/CODE]

sweety439 2022-07-05 12:36

Some families cannot have covering sets (covering congruence, algebraic factorization, or combine of them) and thus there must be a prime of this form.

For the standard notation of family: (a*b^n+c)/gcd(a+c,b-1) (a >= 1, b >= 2 (b is the base), c != 0, gcd(a,c) = 1, gcd(b,c) = 1), for this minimal prime (start with b+1) problem, we require a lower bound of n (i.e. n>=n_0 for a given n_0), however, in this research we can let n < n_0, even let n = 0 or n < 0, i.e. n can be any (positive or negative or 0) integer, and take the numerator of the absolute value of (a*b^n+c)/gcd(a+c,b-1) (since if (a*b^n+c)/gcd(a+c,b-1) have covering sets (covering congruence, algebraic factorization, or combine of them), then this covering sets must have a period of n (i.e. depending on (n mod N) for an integer N, where N is the period), even include n = 0 or n < 0, if there is a (positive or negative or 0) integer n such that the numerator of the absolute value of (a*b^n+c)/gcd(a+c,b-1) has no prime factor p not dividing b, nor has algebraic factorization (it has algebraic factorization if and only if there is an integer r>1 such that a*b^n and -c are both r-th powers of rational numbers, or a*b^n*c is perfect 4th power), then (a*b^n+c)/gcd(a+c,b-1) cannot have finite covering sets (covering congruence, algebraic factorization, or combine of them), i.e. if (a*b^n+c)/gcd(a+c,b-1) has covering sets, then the covering sets must be infinite, and thus there must be a prime of this form

e.g. the form D{B} in base 16, its formula is (206*16^n-11)/15, and we have (take the numerator of the absolute value of the numbers): (although for this minimal prime (start with b+1) problem, n must be >= 1, but in this research of covering sets, we extend to all (positive or negative or 0) integer n

n = 2: 5*19*37
n = 1: 3 * 73
n = 0: 13
n = -1: 1
n = -2: 3*29

Since for n = -1, its value is 1, and 1 has no prime factors, besides, there is no integer r>1 such that 206*16^(-1) and 11 are both r-th powers of rational numbers, and 206*16^(-1)*(-11) is not 4th power of rational number, thus the form D{B} in base 16 cannot have any kinds of covering sets, and there must be a prime of this form.

The form 4{D} in base 14, its formula is 5*14^n-1

n = 2: 11 * 89
n = 1: 3 * 23
n = 0: 2^2
n = -1: 3^2
n = -2: 191

Since for n = 0, its value is 4, and the only prime factor of 4 is 2, but since 2 divides 14 and then 2 cannot appear in the covering set of 5*14^n-1, besides, there is no integer r>1 such that 5*14^0 and 1 are both r-th powers of rational numbers, and 5*14^0*(-1) is not 4th power of rational number, thus the form 4{D} in base 14 cannot have any kinds of covering sets, and there must be a prime of this form.

The form 9{6}M in base 25, its formula is (37*25^n+63)/4

n = 2: 11 * 17 * 31
n = 1: 13 * 19
n = 0: 5^2
n = -1: 13 * 31
n = -2: 59 * 167

Since for n = 0, its value is 25, and the only prime factor of 25 is 5, but since 5 divides 25 and then 5 cannot appear in the covering set of (37*25^n+63)/4, besides, there is no integer r>1 such that 37*25^0 and -63 are both r-th powers of rational numbers, and 37*25^0*63 is not 4th power of rational number, thus the form 9{6}M in base 25 cannot have any kinds of covering sets, and there must be a prime of this form.

Note: 4*72^n-1 does not apply this, since although 4*72^0-1 has no prime factor p not dividing 72, but 4*72^0 and 1 are both squares, thus it has algebraic factorization of difference of squares, thus we cannot show that 4*72^n-1 has no covering set.

An interesting case, there is [B]two[/B] such k but still no easy prime:

Base 312, form C{0}1, its formula is 12*312^n+1:

n = 2: 229 * 5101
n = 1: 5 * 7 * 107
n = 0: 13
n = -1: 3^3
n = -2: 7 * 19 * 61

For n = 0, its value is 13, and the only prime factor of 13 is 13, but since 13 divides 312 and 13 cannot appear in the coveting set of 12*312^n+1

For n = -1, its value is 27, and the only prime factor of 27 is 3, but since 3 divides 312 and 3 cannot appear in the coveting set of 12*312^n+1

And 12*312^n+1 has no algebraic factorization for any n (since the difference of the exponents of 3 and the exponents of 13 is 1, and thus there cannot be r>1 dividing both of them, thus 12*312^n is never perfect power nor of the form 4*m^4), however, 12*312^n+1 has no easy prime, the first prime is n=21162

Base 100, family 4{3}, its formula is (133*100^n-1)/33

n = 2: 41 * 983
n = 1: 13 * 31
n = 0: 2^2
n = -1: 1
n = -2: 13 * 23

For n = 0, its value is 4, and the only prime factor of 4 is 2, but since 2 divides 100 and 2 cannot appear in the coveting set of (133*100^n-1)/33

For n = -1, its value is 1, and 1 has no prime factors

And (133*100^n-1)/33 has no algebraic factorization for any n (since the exponents of 7 and 19 are both 1, thus 133*100^n is never perfect power nor of the form 4*m^4), however, (133*100^n-1)/33 has no easy prime, the first prime is n=5496

However, family 5{7} in base 11 does not apply this (there is no (positive or negative or 0) integer n such that (57*11^n-7)/10 has no prime factor p not dividing 11), thus we cannot show that 5{7} in base 11 has no covering set.

An interesting case is (2*b+1)*b^n-1 (for even b), and its dual form b^n-(2*b+1) (for even b), it applies this (by choose n = 0 for both (2*b+1)*b^n-1 and b^n-(2*b+1), both will get 2*b, and since b is even, it cannot have prime factors not dividing b, but since all prime factors of b+1 divides 1/2 of numbers in these two sequences (for all odd b, both (2*b+1)*b^n-1 and b^n-(2*b+1) are divisible by b+1), thus its Nash weight must * (1/2) and become very low (usually, low-weight (a*b^n+c)/gcd(a+c,b-1) (a >= 1, b >= 2 (b is the base), c != 0, gcd(a,c) = 1, gcd(b,c) = 1) forms have a prime factor p dividing (a*b^n+c)/gcd(a+c,b-1) (a >= 1, b >= 2 (b is the base), c != 0, gcd(a,c) = 1, gcd(b,c) = 1) for either all even n or all odd n) and may have no easy prime.

If 2*b+1 is perfect square, then these two forms have a covering set with combine of covering congruence (one-cover with b+1) and algebraic factorization (difference-of-two-squares factorization)

For the original form:

b = 46 is a classic example, its smallest prime is 93*46^24162-1

For the dual form:

b = 18 has smallest prime 18^768-37, and this prime is a minimal prime (start with b+1) in this base (the smallest prime of the dual form (if exists) must be a minimal prime (start with b+1) in this base if b is divisible by 6)

A similar example is (b+2)*b^n-1 with even b, it applies this (by choose n = -1), but since all prime factors of b+1 divides 1/2 of numbers, it have no easy prime for b = 352, 430, and many bases b, but it cannot produce minimal primes (start with b+1)

sweety439 2022-07-06 13:28

[QUOTE=sweety439;606772]Records for the lengths of the numbers in these families in base b:

{1}: ((b^n-1)/(b-1), length n)

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)

1{0}1: (b^n+1, length n+1)

2 (2)
14 (3)
34 (5)

1{0}2: (b^n+2, length n+1)

3 (2)
23 (12)
47 (114)
89 (256)

{z}y: (b^n-2, length n)

3 (2)
11 (4)
17 (6)
23 (24)
79 (38)
81 (130)
97 (747)

1{0}3: (b^n+3, length n+1)

4 (2)
22 (3)
32 (4)
46 (21)
292 (40)
382 (256)

{z}x: (b^n-3, length n)

4 (2)
16 (3)
22 (6)
28 (10)
50 (21)
52 (105)
94 (204)
152 (346)
154 (396)

1{0}4: (b^n+4, length n+1)

5 (3)
23 (7)

{z}w: (b^n-4, length n)

5 (5)
27 (7)
35 (13)
47 (65)
65 (175)
123 (299)
141 (395)

1{0}z: (b^n+(b-1), length n+1)

2 (2)
5 (3)
14 (17)
32 (109)
80 (195)

{z}1: (b^n-(b-1), length n)

2 (2)
5 (5)
8 (13)
20 (17)
29 (33)
37 (67)

1{0}11: (b^n+(b+1), length n+1)

2 (3)
9 (4)
11 (5)
23 (10)
35 (16)
63 (74)
68 (596)

{z}yz: (b^n-(b+1), length n)

2 (3)
13 (4)
19 (5)
33 (7)
37 (9)
43 (31)
52 (108)
99 (131)
190 (562)
213 (643)

2{0}1: (2*b^n+1, length n+1)

3 (2)
12 (4)
17 (48)
38 (2730)

1{z}: (2*b^n-1, length n+1)

2 (2)
5 (5)
20 (11)
29 (137)
67 (769)

3{0}1: (3*b^n+1, length n+1)

4 (2)
8 (3)
18 (4)
28 (8)
44 (10)
62 (13)
72 (15)
108 (271)
314 (281)

2{z}: (3*b^n-1, length n+1)

4 (2)
12 (3)
32 (12)
42 (2524)

4{0}1: (4*b^n+1, length n+1)

5 (3)
17 (7)
23 (343)

3{z}: (4*b^n-1, length n+1)

5 (2)
23 (6)
47 (1556)

z{0}1: ((b-1)*b^n+1, length n+1)

2 (2)
5 (3)
10 (4)
11 (11)
19 (30)
41 (81)
53 (961)

y{z}: ((b-1)*b^n-1, length n+1)

2 (3)
8 (4)
15 (15)
23 (56)
26 (134)

11{0}1: ((b+1)*b^n+1, length n+2)

2 (3)
9 (4)
18 (11)
51 (185)
63 (187)
108 (400)

10{z}: ((b+1)*b^n-1, length n+2)

2 (3)
12 (4)
17 (6)
23 (9)
42 (10)

{y}z: (((b-2)*b^n+1)/(b-1), length n)

3 (1)
5 (2)
13 (564)
83 (680)[/QUOTE]

An interesting thing:

b = 20 and b = 23, family z{0}1 both have length 15
b = 20 and b = 23, family {z}1 both have length 17

sweety439 2022-07-07 03:48

This minimal prime (start with b+1) problem in base b = 100:

* Including 12:11^68 and 45:11^386 and 56:11^9235 and 67:11^105 (see [URL="https://oeis.org/A069568"]https://oeis.org/A069568[/URL])
* Including 2:22^9576:99 (see [URL="https://oeis.org/A111056"]https://oeis.org/A111056[/URL])
* Including 64:0^529396:1 and 75:0^16391:1 (see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S100"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S100[/URL])
* Including 73:99^44709 (see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R100"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R100[/URL])
* Including 4:3^5496 (see [URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub[/URL])
* Including 52:11^4451 (see [URL="https://stdkmd.net/nrr/prime/primecount.txt"]https://stdkmd.net/nrr/prime/primecount.txt[/URL])
* The unsolved families 3:{43} and 7:{17} (see [URL="http://www.worldofnumbers.com/undulat.htm"]http://www.worldofnumbers.com/undulat.htm[/URL])

But not including 7:11^5452:17 (see [URL="https://stdkmd.net/nrr/7/71117.htm"]https://stdkmd.net/nrr/7/71117.htm[/URL]), since both 7:{11} and {11}:17 have small primes.
But not including 1:11^356:33 (see [URL="https://stdkmd.net/nrr/prime/primecount.txt"]https://stdkmd.net/nrr/prime/primecount.txt[/URL]), since this prime is covered by 1:11^9

Note: 87:{11} is not unsolved family, since it can be ruled out as only contain composites, (784*100^n-1)/9 = ((28*10^n-1)/9) * (28*10^n+1)
Note: 38:{11} is not unsolved family, since it can be ruled out as only contain composites (the proof is more complex, it is combine of difference-of-cubes and two small prime factors (3 and 37))

sweety439 2022-07-11 07:48

Two unusual things:

For the number of minimal primes: (This situation will not occur in larger bases)

* This new problem (i.e. prime > base is needed)

Base 9 (151) > Base 12 (106) > Base 10 (77) > Base 8 (75)

* Original problem (i.e. prime > base is not needed)

Base 10 (26) > Base 12 (17) > Base 8 (15) > Base 9 (12)

The bases which are completely solved but with large minimal primes (start with b+1) in some classic simple families:

* Base 7: {3}1 (indeed corresponding to the largest minimal prime (start with b+1))
* Base 8: {z}1
* Base 11: {1}, z{0}1
* Base 14: 1{0}z, 4{z} (indeed, family 4{z} corresponding to the largest minimal prime (start with b+1))
* Base 15: {3}1, y{z}
* Base 20: 1{0}7, 1{z}, z{0}1, {z}1
* Base 24: 5{0}1, y{z}

For unsolved bases such as:

* Base 13: {y}z
* Base 17: 2{0}1
* Base 19: z{0}1
* Base 28: {z}x

The most unusual is for the original minimal prime problem (i.e. prime > base is not needed) in base 23, which is already solved (if probable primes are allowed), but these families all have large minimal primes:

4{0}1, y{z}, z{0}1, {z}1, {z}y (in fact, also 1{0}2, but 1{0}2 produces minimal prime only in this new problem (i.e. prime > base is needed)

Finally, base 7 is much easier in base 9 (and all bases > 7), for both “number of minimal primes (start with b+1)” and “the largest minimal prime (start with b+1)”, base 7 is both much less than base 9, the largest minimal prime (start with b+1) in base 7 has only length 17, less than all bases except 2, 3, 4, 6, but for both the smallest GFN prime and the smallest GRU prime, base 7 sets records: 3334 and 11111, with lengths 4 and 5, larger than all other bases <= 10

sweety439 2022-07-12 02:16

Family x{y} always produce minimal primes (start with b+1), unless y=1
Family {x}y always produce minimal primes (start with b+1), unless x=1
Family xy{x} always produce minimal primes (start with b+1) if there is no possible prime of the form y{x}, unless x=1
Family {x}yx always produce minimal primes (start with b+1) if there is no possible prime of the form {x}y, unless x=1

All minimal prime (start with b’+1) in base b’=b^n with integer n is always minimal prime (start with b+1) in base b, if this prime is > b

Family A{1} in base 22 cannot produce minimal prime (start with b+1) in base b=22, since its repeating digits is indeed 1, however, it can produce minimal prime (start with b+1) in base b=484=22^2, since we can separate this family to:

A:{11} in base 484
A1:{11} in base 484

And both of them can produce minimal prime (start with b+1) in base b=484, and both of them have prime candidates

Family 19{1} in base 11 cannot produce minimal prime (start with b+1) in base b=11, neither can in base b=121=11^2, since:

1:91:{11} in base 121 ——> covered by the prime 1:11^8, thus cannot produce minimal prime (start with b+1)
19:{11} in base 121 ——> always divisible by 2 and cannot be prime

However, it can produce minimal prime (start with b+1) in base b=1331=11^3, since we can separate this family to:

1:911:{111} in base 1331 ——> covered by the prime 1:111^6, thus cannot produce minimal prime (start with b+1), but this situation does not exist since it is always divisible by 19 and cannot be prime
19:{111} in base 1331
191:{111} in base 1331

And both the second and third of them can produce minimal prime (start with b+1) in base b=1331, and both the second and third of them have prime candidates

sweety439 2022-07-14 00:27

Try to find the minimal sets (in decimal (base 10)) of "{primes} intersection {numbers > 10} intersection {numbers == 1 mod 4}" and "{primes} intersection {numbers > 10} intersection {numbers == 3 mod 4}" (see [URL="https://mersenneforum.org/showpost.php?p=572226&postcount=123"]this post[/URL])

For the numbers == 1 mod 4 case, since the numbers not containing the digit 5 is already covered by [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/primes1mod4/minimal.10.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/primes1mod4/minimal.10.txt[/URL], thus we may assume the number contains at least one 5

Since the number N is prime and > 10 and == 1 mod 4, we can consider the following possible final two digits: (note: we already assume that N contains at least one 5)

Case 1: Final two digits is 01:

then we can write N = x5y01

x cannot contain 1, or 101 ◁ N
x cannot contain 4, or 41 ◁ N
x cannot contain 5, or [B]5501[/B] ◁ N
x cannot contain 6, or 61 ◁ N
x cannot contain 7, or 701 ◁ N
x cannot contain 8, or [B]8501[/B] ◁ N

y cannot contain 1, or 101 ◁ N
y cannot contain 2, or 521 ◁ N
y cannot contain 3, or 53 ◁ N
y cannot contain 4, or 41 ◁ N
y cannot contain 5, or [B]5501[/B] ◁ N
y cannot contain 6, or 61 ◁ N
y cannot contain 7, or 701 ◁ N
y cannot contain 8, or [B]5801[/B] ◁ N

thus we may assume N ∈ {2,3,9}{0,2,3,9}5{0,9}01

If x does not contain 2, then N ∈ {3,9}{0,3,9}5{0,9}01, and hence N is divisible by 3 and cannot be prime, on the other hand, if x contains at least two 2's, then [B]22501[/B] ◁ N, thus we may assume x contains exactly one 2

thus we may assume N ∈ r2s5y01, where r,s ∈ {0,3,9} and y ∈ {0,9}

If y contains 9, then 29 ◁ N, thus we may assume y ∈ {0}

If s contains 9, then 29 ◁ N, thus we may assume s ∈ {0,3}

If s contains at least two 3's, then 3301 ◁ N, thus we may assume s ∈ {0} union {0}3{0}

Thus we have N ∈ {3,9}{0,3,9}2{0}5{0}01 union {3,9}{0,3,9}2{0}3{0}5{0}01

* If s or y (or both) contains 0, then if r contains 3 or 9, then (respectively) 3001 ◁ N or 9001 ◁ N, since r cannot begin with 0, then we may assume r is empty, and hence N ∈ {2{0}5{0}01, 2{0}3{0}5{0}01}

** For the family 2{0}5{0}01, we found the prime [B]2005001[/B], and for the remaining families:

*** For 25{0}1, we found the smallest prime [B]2500000001[/B]

*** For 205{0}1, we found the smallest prime [B]205000001[/B]

*** For 2{0}501, we found the smallest prime [B]2000000000000501[/B]

* If neither s nor y contains 0, then we may assume N ∈ {3,9}{0,3,9}2501 union {3,9}{0,3,9}23501

** If r contains 0, then r must contains subsequence 30 or 90, then we must have 3001 ◁ N or 9001 ◁ N, thus we may assume N ∈ {3,9}2501 union {3,9}23501

*** For the {3,9}2501 case, if r contains at least two 3's, then 3301 ◁ N, if r contains at least two 9's, then 9901 ◁ N, thus we may assume N ∈ {2501,32501,92501,392501,932501}, however, none of these numbers is prime

*** For the {3,9}23501 case, if r contains 3, then 3301 ◁ N, if r contains 9, then [B]923501[/B] ◁ N, thus we may assume r is empty, and hence N = 23501, but 23501 is not prime

Case 2: Final two digits is 09:

Then [B]509[/B] ◁ N

Case 3: Final two digits is 13:

Then 13 ◁ N

Case 4: Final two digits is 17:

Then 17 ◁ N

Case 5: Final two digits is 21:

Then [B]521[/B] ◁ N

Case 6: Final two digits is 29:

Then 29 ◁ N

Case 7: Final two digits is 33:

Then 53 ◁ N

Case 8: Final two digits is 37:

Then 37 ◁ N

Case 9: Final two digits is 41:

Then 41 ◁ N

Case 10: Final two digits is 49:

then we can write N = x5y49

x cannot contain 1, or 149 ◁ N
x cannot contain 2, or 29 ◁ N
x cannot contain 3, or 349 ◁ N
x cannot contain 4, or 449 ◁ N
x cannot contain 7, or [B]7549[/B] ◁ N
x cannot contain 8, or 89 ◁ N

y cannot contain 1, or 149 ◁ N
y cannot contain 2, or 29 ◁ N
y cannot contain 3, or 349 ◁ N
y cannot contain 4, or 449 ◁ N
y cannot contain 7, or [B]5749[/B] ◁ N
y cannot contain 8, or 89 ◁ N

sweety439 2022-07-14 03:17

If we write all minimal primes (start with b+1) in base b one time, then we will write these numbers of digits:

[CODE]
b=2:

0 0's
2 1's

b=3:

0 0's
5 1's
2 2's

b=4:

0 0's
5 1's
3 2's
3 3's

b=5:

105 0's
24 1's
4 2's
22 3's
14 4's

b=6:

3 0's
9 1's
2 2's
2 3's
8 4's
5 5's

b=7:

43 0's
73 1's
21 2's
67 3's
24 4's
51 5's
9 6's

b=8:

22 0's
49 1's
13 2's
24 3's
283 4's
48 5's
25 6's
59 7's

b=9:

1350 0's
174 1's
32 2's
108 3's
24 4's
107 5's
357 6's
794 7's
58 8's

b=10:

69 0's
33 1's
27 2's
9 3's
19 4's
45 5's
21 6's
31 7's
22 8's
34 9's

b=11:

2666 0's
523 1's
227 2's
250 3's
722 4's
1514 5's (unless 5(7^62668) is in fact composite and there is no prime of the form 5{7}, in this case there are 1513 5's, but this is very impossible)
251 6's
66917 7's (unless 5(7^62668) is in fact composite, in this case there are n+4249 (must be > 66917) 7's where n is the smallest number n such that 5(7^n) is prime (must be > 62668) if there is a prime of the form 5{7}, or there are 4249 7's if there is no prime of the form 5{7}, but 5(7^62668) is in fact composite is very impossible)
357 8's
592 9's
1395 A's

b=12:

105 0's
42 1's
28 2's
4 3's
25 4's
23 5's
14 6's
43 7's
4 8's
38 9's
38 A's
69 B's
[/CODE]

sweety439 2022-07-14 03:40

[QUOTE=sweety439;571906]

[CODE]
b=2, d=0: 0
b=2, d=1: 2 (the prime 11)
b=3, d=0: 0
b=3, d=1: 3 (the prime 111)
b=3, d=2: 1 (the primes 12 and 21)
b=4, d=0: 0
b=4, d=1: 2 (the prime 11)
b=4, d=2: 2 (the prime 221)
b=4, d=3: 1 (the primes 13, 23, 31)
b=5, d=0: 93 (the prime 10[SUB]93[/SUB]13)
b=5, d=1: 3 (the prime 111)
b=5, d=2: 1 (the primes 12, 21, 23, 32)
b=5, d=3: 4 (the prime 33331)
b=5, d=4: 4 (the primes 14444 and 44441)
b=6, d=0: 2 (the prime 40041)
b=6, d=1: 2 (the prime 11)
b=6, d=2: 1 (the primes 21 and 25)
b=6, d=3: 1 (the primes 31 and 35)
b=6, d=4: 3 (the prime 4441)
b=6, d=5: 1 (the primes 15, 25, 35, 45, 51)
b=7, d=0: 7 (the prime 5100000001)
b=7, d=1: 5 (the prime 11111)
b=7, d=2: 3 (the prime 1222)
b=7, d=3: 16 (the prime 3[SUB]16[/SUB]1)
b=7, d=4: 2 (the primes 344, 445, 544, 4504, 40054)
b=7, d=5: 4 (the prime 35555)
b=7, d=6: 2 (the prime 6634)
b=8, d=0: 3 (the prime 500025)
b=8, d=1: 3 (the prime 111)
b=8, d=2: 2 (the prime 225)
b=8, d=3: 3 (the prime 3331)
b=8, d=4: 220 (the prime 4[SUB]220[/SUB]7)
b=8, d=5: 14 (the prime 5[SUB]13[/SUB]25)
b=8, d=6: 2 (the primes 661 and 667)
b=8, d=7: 12 (the prime 7[SUB]12[/SUB]1)
b=9, d=0: 1158 (the prime 30[SUB]1158[/SUB]11)
b=9, d=1: 36 (the prime 561[SUB]36[/SUB])
b=9, d=2: 4 (the prime 22227)
b=9, d=3: 8 (the prime 8333333335)
b=9, d=4: 11 (the prime 54[SUB]11[/SUB])
b=9, d=5: 4 (the prime 55551)
b=9, d=6: 329 (the prime 76[SUB]329[/SUB]2)
b=9, d=7: 687 (the prime 27[SUB]686[/SUB]07)
b=9, d=8: 19 (the prime 8[SUB]19[/SUB]335)
b=10, d=0: 28 (the prime 50[SUB]28[/SUB]27)
b=10, d=1: 2 (the prime 11)
b=10, d=2: 3 (the prime 2221)
b=10, d=3: 1 (the primes 13, 23, 31, 37, 43, 53, 73, 83, 349)
b=10, d=4: 2 (the prime 449)
b=10, d=5: 11 (the prime 5[SUB]11[/SUB]1)
b=10, d=6: 4 (the prime 666649)
b=10, d=7: 2 (the primes 277, 577, 727, 757, 787, 877)
b=10, d=8: 2 (the prime 881)
b=10, d=9: 3 (the prime 9949)
b=11, d=0: 126 (the prime 50[SUB]126[/SUB]57)
b=11, d=1: 17 (the prime 1[SUB]17[/SUB])
b=11, d=2: 6 (the prime 5222222)
b=11, d=3: 10 (the prime 3[SUB]10[/SUB]7)
b=11, d=4: 44 (the prime 4[SUB]44[/SUB]1)
b=11, d=5: 221 (the prime 85[SUB]220[/SUB]05]
b=11, d=6: 124 (the prime 326[SUB]124[/SUB])
b=11, d=7: 62668 (the prime 57[SUB]62668[/SUB])
b=11, d=8: 17 (the prime 8[SUB]17[/SUB]3)
b=11, d=9: 32 (the prime 9[SUB]32[/SUB]1)
b=11, d=A: 713 (the prime A[SUB]713[/SUB]58)
b=12, d=0: 39 (the prime 40[SUB]39[/SUB]77)
b=12, d=1: 2 (the prime 11)
b=12, d=2: 3 (the prime 222B)
b=12, d=3: 1 (the primes 31, 35, 37, 3B)
b=12, d=4: 3 (the prime 4441)
b=12, d=5: 2 (the primes 565 and 655)
b=12, d=6: 2 (the prime 665)
b=12, d=7: 3 (the primes 4777 and 9777)
b=12, d=8: 1 (the primes 81, 85, 87, 8B)
b=12, d=9: 4 (the prime 9999B)
b=12, d=A: 4 (the prime AAAA1)
b=12, d=B: 7 (the prime BBBBBB99B)
b=13, d=0: 32017 (the prime 80[SUB]32017[/SUB]111)
b=13, d=1: 5 (the prime 11111)
b=13, d=2: 77 (the prime 72[SUB]77[/SUB])
b=13, d=3: >82000 (the prime A3[SUB]n[/SUB]A)
b=13, d=4: 14 (the prime 94[SUB]14[/SUB])
b=13, d=5: >88000 (the prime 95[SUB]n[/SUB])
b=13, d=6: 137 (the prime 6[SUB]137[/SUB]A3)
b=13, d=7: 1504 (the prime 7[SUB]1504[/SUB]1)
b=13, d=8: 53 (the prime 8[SUB]53[/SUB]7)
b=13, d=9: 1362 (the prime 9[SUB]1362[/SUB]5)
b=13, d=A: 95 (the prime C5A[SUB]95[/SUB])
b=13, d=B: 834 (the prime B[SUB]834[/SUB]74)
b=13, d=C: 10631 (the prime C[SUB]10631[/SUB]92)
b=14, d=0: 83 (the prime 40[SUB]83[/SUB]49)
b=14, d=1: 3 (the prime 111)
b=14, d=2: 3 (the prime B2225)
b=14, d=3: 5 (the prime A33333)
b=14, d=4: 63 (the prime 4[SUB]63[/SUB]09)
b=14, d=5: 36 (the prime 85[SUB]36[/SUB])
b=14, d=6: 10 (the prime 86[SUB]10[/SUB]99)
b=14, d=7: 2 (the primes 771, 77D)
b=14, d=8: 86 (the prime 8[SUB]86[/SUB]B)
b=14, d=9: 37 (the prime 9[SUB]36[/SUB]89)
b=14, d=A: 59 (the prime A[SUB]59[/SUB]3)
b=14, d=B: 78 (the prime 6B[SUB]77[/SUB]2B)
b=14, d=C: 79 (the prime 8C[SUB]79[/SUB]3)
b=14, d=D: 19698 (the prime 4D[SUB]19698[/SUB])
b=15, d=0: 33 (the prime 50[SUB]33[/SUB]17)
b=15, d=1: 3 (the prime 111)
b=15, d=2: 9 (the prime 2222222252)
b=15, d=3: 12 (the prime 3[SUB]12[/SUB]1)
b=15, d=4: 3 (the prime 4434)
b=15, d=5: 8 (the prime 555555557)
b=15, d=6: 104 (the prime 96[SUB]104[/SUB]08)
b=15, d=7: 156 (the prime 7[SUB]155[/SUB]97)
b=15, d=8: 8 (the prime 8888888834)
b=15, d=9: 10 (the prime 9999999999D)
b=15, d=A: 4 (the prime AAAA52)
b=15, d=B: 31 (the prime EB[SUB]31[/SUB])
b=15, d=C: 10 (the prime DCCCCCCCCCC8)
b=15, d=D: 16 (the prime D[SUB]16[/SUB]B)
b=15, d=E: 145 (the prime E[SUB]145[/SUB]397)
b=16, d=0: 3542 (the prime 90[SUB]3542[/SUB]91)
b=16, d=1: 2 (the prime 11)
b=16, d=2: 32 (the prime 2[SUB]32[/SUB]7)
b=16, d=3: >76000 (the prime 3[SUB]n[/SUB]AF)
b=16, d=4: 72785 (the prime 4[SUB]72785[/SUB]DD)
b=16, d=5: 70 (the prime A015[SUB]70[/SUB])
b=16, d=6: 87 (the prime 56[SUB]87[/SUB]F)
b=16, d=7: 20 (the prime 7[SUB]19[/SUB]87)
b=16, d=8: 1517 (the prime F8[SUB]1517[/SUB]F)
b=16, d=9: 1052 (the prime D9[SUB]1052[/SUB])
b=16, d=A: 305 (the prime DA[SUB]305[/SUB]5)
b=16, d=B: 32234 (the prime DB[SUB]32234[/SUB])
b=16, d=C: 3700 (the prime 5BC[SUB]3700[/SUB]D)
b=16, d=D: 39 (the prime 4D[SUB]39[/SUB])
b=16, d=E: 34 (the prime E[SUB]34[/SUB]B)
b=16, d=F: 1961 (the prime 300F[SUB]1960[/SUB]AF)
[/CODE][/QUOTE]

Some of them are conjectured numbers (they are 99.9999999999999... (> 10000 9's) correct, but not 100% correct), they are:

b=11, d=7:

* If 5(7^62668) is in fact composite but there is a larger prime 5{7}, then it will be the smallest n such that 5(7^n) is prime, and must be > 62668
* If 5(7^62668) is in fact composite and there is no prime of the form 5{7}, then it is 1011 (the prime 557[SUB]1011[/SUB])

b=13, d=0:

* If 8(0^32017)111 is in fact composite but there is a larger prime 8{0}111, then it will be the smallest n such that 8(0^n)111 is prime, and must be > 32017
* If 8(0^32017)111 is in fact composite and there is no prime of the form 8{0}111, then it is 6540 (the prime B0[SUB]6540[/SUB]BBA)

b=13, d=3:

* If there is no prime of the form A{3}A, then it is 178 (the prime 3[SUB]178[/SUB]5)

b=13, d=5:

* If there is no prime of the form 9{5} and C(5^23755)C is in fact prime, then it is 23755
* If there is no prime of the form 9{5} and C(5^23755)C is in fact composite but there is a larger prime C{5}C, then it will be the smallest n such that C(5^n)C is prime, and must be > 23755
* If there is no prime of the form 9{5} and C(5^23755)C is in fact composite and there is no prime of the form C{5}C, then it is 713 (the prime CC5[SUB]713[/SUB])

b=16, d=3:

* If there is no prime of the form {3}AF, then it is 24 (the prime 3[SUB]24[/SUB]1)

b=16, d=4:

* If (4^72785)DD is in fact composite but there is a larger prime {4}DD, then it will be the smallest n such that (4^n)DD is prime, and must be > 72785
* If (4^72785)DD is in fact composite and there is no prime of the form {4}DD, then it is 263 (the prime D4[SUB]263[/SUB]D)

b=16, d=B:

* If D(B^32234) is in fact composite but there is a larger prime D{B}, then it will be the smallest n such that D(B^n) is prime, and must be > 32234
* If D(B^32234) is in fact composite and there is no prime of the form D{B}, then it is 17804 (the prime D0B[SUB]17804[/SUB])

sweety439 2022-07-17 16:01

[QUOTE=sweety439;609479]If we write all minimal primes (start with b+1) in base b one time, then we will write these numbers of digits:

[CODE]
b=2:

0 0's
2 1's

b=3:

0 0's
5 1's
2 2's

b=4:

0 0's
5 1's
3 2's
3 3's

b=5:

105 0's
24 1's
4 2's
22 3's
14 4's

b=6:

3 0's
9 1's
2 2's
2 3's
8 4's
5 5's

b=7:

43 0's
73 1's
21 2's
67 3's
24 4's
51 5's
9 6's

b=8:

22 0's
49 1's
13 2's
24 3's
283 4's
48 5's
25 6's
59 7's

b=9:

1350 0's
174 1's
32 2's
108 3's
24 4's
107 5's
357 6's
794 7's
58 8's

b=10:

69 0's
33 1's
27 2's
9 3's
19 4's
45 5's
21 6's
31 7's
22 8's
34 9's

b=11:

2666 0's
523 1's
227 2's
250 3's
722 4's
1514 5's (unless 5(7^62668) is in fact composite and there is no prime of the form 5{7}, in this case there are 1513 5's, but this is very impossible)
251 6's
66917 7's (unless 5(7^62668) is in fact composite, in this case there are n+4249 (must be > 66917) 7's where n is the smallest number n such that 5(7^n) is prime (must be > 62668) if there is a prime of the form 5{7}, or there are 4249 7's if there is no prime of the form 5{7}, but 5(7^62668) is in fact composite is very impossible)
357 8's
592 9's
1395 A's

b=12:

105 0's
42 1's
28 2's
4 3's
25 4's
23 5's
14 6's
43 7's
4 8's
38 9's
38 A's
69 B's
[/CODE][/QUOTE]

Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, [URL="http://factordb.com/index.php?id=3"]sum[/URL], [URL="http://factordb.com/index.php?id=3"]product[/URL]

Base 3:

3 primes, totally 7 digits, [URL="http://factordb.com/index.php?id=25"]sum[/URL], [URL="http://factordb.com/index.php?id=455"]product[/URL]

Base 4:

5 primes, totally 11 digits, [URL="http://factordb.com/index.php?id=77"]sum[/URL], [URL="http://factordb.com/index.php?id=205205"]product[/URL]

Base 5:

22 primes, totally 169 digits, [URL="http://factordb.com/index.php?id=1100000003799642708"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457822814"]product[/URL]

Base 6:

11 primes, totally 29 digits, [URL="http://factordb.com/index.php?id=7401"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457821560"]product[/URL]

Base 7:

71 primes, totally 288 digits, [URL="http://factordb.com/index.php?id=116315467894207"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457825324"]product[/URL]

Base 8:

75 primes, totally 523 digits, [URL="http://factordb.com/index.php?id=1100000003799644593"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002371473795"]product[/URL]

Base 9:

151 primes, totally 3004 digits, [URL="http://factordb.com/index.php?id=1100000003799645271"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003450366253"]product[/URL]

Base 10:

77 primes, totally 310 digits, [URL="http://factordb.com/index.php?id=1100000003799645582"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002370859491"]product[/URL]

Base 11:

1068 primes, totally 75414 digits, [URL="http://factordb.com/index.php?id=1100000003799646641"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003583737715"]product[/URL]

Base 12:

106 primes, totally 433 digits, [URL="http://factordb.com/index.php?id=1100000003799647067"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457818232"]product[/URL]

Base 14:

650 primes, totally 25404 digits, [URL="http://factordb.com/index.php?id=1100000003799647609"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003575953976"]product[/URL]

Base 15:

1284 primes, totally 8286 digits, [URL="http://factordb.com/index.php?id=1100000003799647942"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003588261354"]product[/URL]

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.

sweety439 2022-07-20 08:03

Two special cases of minimal primes (start with b+1) base b

* lovely numbers base b: Let d_1 and d_2 be digits in base b such that d_1 + d_2 = b, find the smallest prime of the form d_1*b^n+d_2 with n >= 1, this prime is always minimal prime (start with b+1) base b, this includes the special cases d_1 = 1, d_2 = b-1 (which is b^n+(b-1), see [URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]) and d_1 = b-1, d_2 = 1 (which is (b-1)*b^n+1, see [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL] and [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL] and [URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL])
** such prime is always expected to exist as there cannot be covering congruence nor algebraic factorization (since if so, then d_1, d_2, b will be all r-th powers for an odd r > 1, which is impossible (by Fermat Last Theorem)
* flexible numbers base b: Let d be a divisor (>1) of b-1 (if b-1 is prime, then d can only be b-1 itself), find the smallest prime of the form ((d-1)*b^n+1)/d with n >= 2, this prime is always minimal prime (start with b+1) base b
** for the case d = 2, such prime may not exist and it is widely believed that there are only finitely many such primes for fixed base b, since it is generalized half Fermat prime in base b
** for the case d > 2, such prime are usually expected to exist (as there cannot be covering congruence of this form, but there may be algebraic factorization or combine of covering congruence and algebraic factorization if d-1 is indeed perfect odd power (of the form m^r with odd r > 1) or of the form 4*m^4, and if d-1 is of neither of these two forms, then there must be prime of this form), but the smallest such prime may be large, e.g.

[CODE]
b,d,smallest exponent n
13,12,564
17,8,190
23,11,3762
31,6,1026
43,14,580
70,69,555
[/CODE]

sweety439 2022-07-30 02:10

[QUOTE=sweety439;571731]References of given simple families for the minimal primes (start with b+1) problem in bases 2<=b<=1024:

{1}:

[URL="http://www.users.globalnet.co.uk/~aads/primes.html"]http://www.users.globalnet.co.uk/~aads/primes.html[/URL] (broken link: [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]from wayback machine[/URL])
[URL="http://www.users.globalnet.co.uk/~aads/titans.html"]http://www.users.globalnet.co.uk/~aads/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html"]from wayback machine[/URL])
[URL="http://www.primes.viner-steward.org/andy/titans.html"]http://www.primes.viner-steward.org/andy/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html"]from wayback machine[/URL])
[URL="http://www.phi.redgolpe.com/"]http://www.phi.redgolpe.com/[/URL] (broken link: [URL="https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/"]from wayback machine[/URL])
[URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]
[URL="https://oeis.org/A128164/a128164_7.txt"]https://oeis.org/A128164/a128164_7.txt[/URL]
[URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL]
[URL="http://www.mersennewiki.org/index.php/Repunit"]http://www.mersennewiki.org/index.php/Repunit[/URL] (broken link: [URL="https://web.archive.org/web/20180416000002/http://www.mersennewiki.org/index.php/Repunit"]from wayback machine[/URL])
[URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL]
[URL="http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470"]http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470[/URL]
[URL="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906"]https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906[/URL] (archive today cannot automatically return the archive page, if you use archive today, click [URL="https://archive.is/WCvbi"]https://archive.is/WCvbi[/URL])
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://ebisui-hirotaka.com/img/file410.pdf"]http://ebisui-hirotaka.com/img/file410.pdf[/URL]
[URL="https://www.jstor.org/stable/2006470?origin=crossref"]https://www.jstor.org/stable/2006470?origin=crossref[/URL]
[URL="http://www.bitman.name/math/table/379"]http://www.bitman.name/math/table/379[/URL]
[URL="https://oeis.org/A084740"]https://oeis.org/A084740[/URL]
[URL="https://oeis.org/A084738"]https://oeis.org/A084738[/URL] (corresponding primes)
[URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL] (prime bases)
[URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL] (prime bases, corresponding primes)
[URL="https://oeis.org/A128164"]https://oeis.org/A128164[/URL] (length 2 not allowed)
[URL="https://oeis.org/A285642"]https://oeis.org/A285642[/URL] (length 2 not allowed, corresponding primes)

1{0}1:

[URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL]
[URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/index.html"]http://yves.gallot.pagesperso-orange.fr/primes/index.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/stat.html"]http://yves.gallot.pagesperso-orange.fr/primes/stat.html[/URL]
[URL="https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf"]https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf"]https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf"]https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf"]https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf[/URL] (b=2^n)
[URL="https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf"]https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf[/URL] (b=2^n)
[URL="https://arxiv.org/pdf/1605.01371.pdf"]https://arxiv.org/pdf/1605.01371.pdf[/URL] (b=2^n)
[URL="https://oeis.org/A228101"]https://oeis.org/A228101[/URL]
[URL="https://oeis.org/A079706"]https://oeis.org/A079706[/URL]
[URL="https://oeis.org/A084712"]https://oeis.org/A084712[/URL] (corresponding primes)
[URL="https://oeis.org/A123669"]https://oeis.org/A123669[/URL] (length 2 not allowed, corresponding primes)

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1:

[URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL] (2{0}1 in base 512, 4{0}1 in bases 32, 512, 1024, which are not in the first 4 references)
[URL="http://www.prothsearch.com/GFN10.html"]http://www.prothsearch.com/GFN10.html[/URL] (A{0}1 in base 1000, which are not in the first 4 references)
[URL="https://mersenneforum.org/showthread.php?t=6918"]https://mersenneforum.org/showthread.php?t=6918[/URL] (2{0}1)
[URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL] (2{0}1 in bases == 11 mod 12)
[URL="https://oeis.org/A119624"]https://oeis.org/A119624[/URL] (2{0}1)
[URL="https://oeis.org/A253178"]https://oeis.org/A253178[/URL] (2{0}1)
[URL="https://oeis.org/A098872"]https://oeis.org/A098872[/URL] (2{0}1 in bases divisible by 6)
[URL="https://oeis.org/A098877"]https://oeis.org/A098877[/URL] (3{0}1 in bases divisible by 6)
[URL="https://oeis.org/A088782"]https://oeis.org/A088782[/URL] (A{0}1)
[URL="https://oeis.org/A088622"]https://oeis.org/A088622[/URL] (A{0}1, corresponding primes)

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}:

[URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL] (1{z})
[URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL] (1{z})
[URL="https://oeis.org/A119591"]https://oeis.org/A119591[/URL] (1{z})
[URL="https://oeis.org/A098873"]https://oeis.org/A098873[/URL] (1{z} in bases divisible by 6)
[URL="https://oeis.org/A098876"]https://oeis.org/A098876[/URL] (2{z} in bases divisible by 6)

z{0}1:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.prothsearch.com/riesel1a.html"]http://www.prothsearch.com/riesel1a.html[/URL] (base 512)
[URL="http://www.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL]
[URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL] (prime bases)

y{z}:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]https://www.rieselprime.de/ziki/Williams_prime_MM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL]
[URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL]
[URL="http://www.prothsearch.com/riesel2.html"]http://www.prothsearch.com/riesel2.html[/URL] (base 128)
[URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL] (prime bases)

1{0}2:

[URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL]
[URL="https://oeis.org/A084713"]https://oeis.org/A084713[/URL] (corresponding primes)
[URL="https://oeis.org/A138067"]https://oeis.org/A138067[/URL] (length 2 not allowed)

1{0}z:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]
[URL="https://oeis.org/A076846"]https://oeis.org/A076846[/URL] (corresponding primes)
[URL="https://oeis.org/A078178"]https://oeis.org/A078178[/URL] (length 2 not allowed)
[URL="https://oeis.org/A078179"]https://oeis.org/A078179[/URL] (length 2 not allowed, corresponding primes)

{z}1:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/435"]http://www.bitman.name/math/table/435[/URL] (prime bases)
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL]
[URL="https://oeis.org/A343589"]https://oeis.org/A343589[/URL] (corresponding primes)
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL] (prime bases)

11{0}1: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]https://www.rieselprime.de/ziki/Williams_prime_PP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/474"]http://www.bitman.name/math/table/474[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

1{0}11: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A346149"]https://oeis.org/A346149[/URL]
[URL="https://oeis.org/A346154"]https://oeis.org/A346154[/URL] (corresponding primes)

10{z}: (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})

[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_least"]https://www.rieselprime.de/ziki/Williams_prime_PM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_table"]https://www.rieselprime.de/ziki/Williams_prime_PM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/471"]http://www.bitman.name/math/table/471[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

{z}y:

[URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] (length 1 allowed)
[URL="https://oeis.org/A250200"]https://oeis.org/A250200[/URL]
[URL="https://oeis.org/A255707"]https://oeis.org/A255707[/URL] (length 1 allowed)
[URL="https://oeis.org/A084714"]https://oeis.org/A084714[/URL] (length 1 allowed, corresponding primes)
[URL="https://oeis.org/A292201"]https://oeis.org/A292201[/URL] (length 1 allowed, prime bases)

{z}yz: (not minimal prime (start with b+1) if there is smaller prime of the form {z}y)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A178250"]https://oeis.org/A178250[/URL]

{#}$: (for odd base b, # = (b−1)/2, $ = (b+1)/2)

[URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]
[URL="http://www.prothsearch.com/GFN05.html"]http://www.prothsearch.com/GFN05.html[/URL] (base 625)

{z0}z1: (almost cannot be minimal prime (start with b+1), since this is not simple family, but always minimal prime (start with b'+1) in base b'=b^2)

[URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171"]https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf[/URL]
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://www.bitman.name/math/table/488"]http://www.bitman.name/math/table/488[/URL]
[URL="https://oeis.org/A084742"]https://oeis.org/A084742[/URL]
[URL="https://oeis.org/A084741"]https://oeis.org/A084741[/URL] (corresponding primes)
[URL="https://oeis.org/A065507"]https://oeis.org/A065507[/URL] (prime bases)[/QUOTE]

(below, "index" means the index of this prime in the minimal primes (start with b+1) set)

(only list families which [B]must[/B] produce minimal primes (start with b+1))

Family {1}: (not exist for bases in [URL="https://oeis.org/A096059"]https://oeis.org/A096059[/URL])

Base 2: 11, length 2, decimal 3, index 1
Base 3: 111, length 3, decimal 13, index 3
Base 4: 11, length 2, decimal 5, index 1
Base 5: 111, length 3, decimal 31, index 8
Base 6: 11, length 2, decimal 7, index 1
Base 7: 11111, length 5, decimal 2801, index 53
Base 8: 111, length 3, decimal 73, index 16
Base 9: not exist
Base 10: 11, length 2, decimal 11, index 1
Base 11: 11111111111111111, length 17, decimal 50544702849929377, index 975
Base 12: 11, length 2, decimal 13, index 1
Base 13: 11111, length 5, decimal 30941, index 494
Base 14: 111, length 3, decimal 211, index 40
Base 15: 111, length 3, decimal 241, index 43
Base 16: 11, length 2, decimal 17, index 1
Base 17: 111, length 3, decimal 307, index 56
Base 18: 11, length 2, decimal 19, index 1
Base 19: 1111111111111111111, length 19, decimal 109912203092239643840221, index 29382
Base 20: 111, length 3, decimal 421, index 73
Base 21: 111, length 3, decimal 463, index 78
Base 22: 11, length 2, decimal 23, index 1

Family 1{0}1: (not exist for bases == 1 mod 2 and bases in [URL="https://oeis.org/A070265"]https://oeis.org/A070265[/URL])

Base 2: 11, length 2, decimal 3, index 1
Base 3: not exist
Base 4: 11, length 2, decimal 5, index 1
Base 5: not exist
Base 6: 11, length 2, decimal 7, index 1
Base 7: not exist
Base 8: not exist
Base 9: not exist
Base 10: 11, length 2, decimal 11, index 1
Base 11: not exist
Base 12: 11, length 2, decimal 13, index 1
Base 13: not exist
Base 14: 101, length 3, decimal 197, index 39
Base 15: not exist
Base 16: 11, length 2, decimal 17, index 1
Base 17: not exist
Base 18: 11, length 2, decimal 19, index 1
Base 19: not exist
Base 20: 101, length 3, decimal 401, index 71
Base 21: not exist
Base 22: 11, length 2, decimal 23, index 1

Family 2{0}1: (not exist for bases == 1 mod 3)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: 21, length 2, decimal 7, index 2
Base 4: not exist
Base 5: 21, length 2, decimal 11, index 2
Base 6: 21, length 2, decimal 13, index 3
Base 7: not exist
Base 8: 21, length 2, decimal 17, index 3
Base 9: 21, length 2, decimal 19, index 4
Base 10: not exist
Base 11: 21, length 2, decimal 23, index 4
Base 12: 2001, length 4, decimal 3457, index 58
Base 13: not exist
Base 14: 21, length 2, decimal 29, index 4
Base 15: 21, length 2, decimal 31, index 5
Base 16: not exist
Base 17: 200000000000000000000000000000000000000000000001, length 48, decimal 13555929465559461990942712143872578804076607708197374744547, index 10094
Base 18: 21, length 2, decimal 37, index 5
Base 19: not exist
Base 20: 21, length 2, decimal 41, index 5
Base 21: 21, length 2, decimal 43, index 6
Base 22: not exist

Family 1{z}: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 12, length 2, decimal 5, index 1
Base 4: 13, length 2, decimal 7, index 2
Base 5: 14444, length 5, decimal 1249, index 16
Base 6: 15, length 2, decimal 11, index 2
Base 7: 16, length 2, decimal 13, index 2
Base 8: 177, length 3, decimal 127, index 21
Base 9: 18, length 2, decimal 17, index 3
Base 10: 19, length 2, decimal 19, index 4
Base 11: 1AA, length 3, decimal 241, index 37
Base 12: 1B, length 2, decimal 23, index 4
Base 13: 1CC, length 3, decimal 337, index 48
Base 14: 1DDDD, length 5, decimal 76831, index 233
Base 15: 1E, length 2, decimal 29, index 4
Base 16: 1F, length 2, decimal 31, index 5
Base 17: 1GG, length 3, decimal 577, index 86
Base 18: 1HH, length 3, decimal 647, index 66
Base 19: 1I, length 2, decimal 37, index 4
Base 20: 1JJJJJJJJJJ, length 11, decimal 20479999999999, index 3015
Base 21: 1K, length 2, decimal 41, index 5
Base 22: 1L, length 2, decimal 43, index 6

Family 3{0}1: (not exist for bases == 1 mod 2)

Base 2: not interpretable (base 2 has no digit "3")
Base 3: not interpretable (base 3 has no digit "3")
Base 4: 31, length 2, decimal 13, index 4
Base 5: not exist
Base 6: 31, length 2, decimal 19, index 5
Base 7: not exist
Base 8: 301, length 3, decimal 193, index 24
Base 9: not exist
Base 10: 31, length 2, decimal 31, index 7
Base 11: not exist
Base 12: 31, length 2, decimal 37, index 7
Base 13: not exist
Base 14: 31, length 2, decimal 43, index 8
Base 15: not exist
Base 16: 301, length 3, decimal 769, index 69
Base 17: not exist
Base 18: 3001, length 4, decimal 17497, index 195
Base 19: not exist
Base 20: 31, length 2, decimal 61, index 10
Base 21: not exist
Base 22: 31, length 2, decimal 67, index 11

Family 2{z}: (not exist for bases == 1 mod 2)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: not exist
Base 4: 23, length 2, decimal 11, index 3
Base 5: not exist
Base 6: 25, length 2, decimal 17, index 4
Base 7: not exist
Base 8: 27, length 2, decimal 23, index 5
Base 9: not exist
Base 10: 29, length 2, decimal 29, index 6
Base 11: not exist
Base 12: 2BB, length 3, decimal 431, index 34
Base 13: not exist
Base 14: 2D, length 2, decimal 41, index 7
Base 15: not exist
Base 16: 2F, length 2, decimal 47, index 9
Base 17: not exist
Base 18: 2H, length 2, decimal 53, index 9
Base 19: not exist
Base 20: 2J, length 2, decimal 59, index 9
Base 21: not exist
Base 22: 2LL, length 3, decimal 1451, index 127

Family 4{0}1: (not exist for bases == 1 mod 5 and bases == 14 mod 15 and bases which are 4th powers)

Base 2: not interpretable (base 2 has no digit "4")
Base 3: not interpretable (base 3 has no digit "4")
Base 4: not interpretable (base 4 has no digit "4")
Base 5: 401, length 3, decimal 101, index 12
Base 6: not exist
Base 7: 41, length 2, decimal 29, index 6
Base 8: 401, length 3, decimal 257, index 27
Base 9: 41, length 2, decimal 37, index 8
Base 10: 41, length 2, decimal 41, index 9
Base 11: not exist
Base 12: 401, length 3, decimal 577, index 35
Base 13: 41, length 2, decimal 53, index 10
Base 14: not exist
Base 15: 41, length 2, decimal 61, index 12
Base 16: not exist
Base 17: 4000001, length 7, decimal 96550277, index 5138
Base 18: 41, length 2, decimal 73, index 14
Base 19: 4001, length 4, decimal 27437, index 748
Base 20: 401, length 3, decimal 1601, index 120
Base 21: not exist
Base 22: 41, length 2, decimal 89, index 16

Family 3{z}: (not exist for bases == 1 mod 3 and bases == 4 mod 5 and bases which are squares)

Base 2: not interpretable (base 2 has no digit "3")
Base 3: not interpretable (base 3 has no digit "3")
Base 4: not exist
Base 5: 34, length 2, decimal 19, index 5
Base 6: 35, length 2, decimal 23, index 6
Base 7: not exist
Base 8: 37, length 2, decimal 31, index 7
Base 9: not exist
Base 10: not exist
Base 11: 3A, length 2, decimal 43, index 9
Base 12: 3B, length 2, decimal 47, index 10
Base 13: not exist
Base 14: not exist
Base 15: 3E, length 2, decimal 59, index 11
Base 16: not exist
Base 17: 3G, length 2, decimal 67, index 12
Base 18: 3H, length 2, decimal 71, index 13
Base 19: not exist
Base 20: 3J, length 2, decimal 79, index 14
Base 21: 3K, length 2, decimal 83, index 15
Base 22: not exist

Family z{0}1: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 21, length 2, decimal 7, index 2
Base 4: 31, length 2, decimal 13, index 4
Base 5: 401, length 3, decimal 101, index 12
Base 6: 51, length 2, decimal 31, index 8
Base 7: 61, length 2, decimal 43, index 10
Base 8: 701, length 3, decimal 449, index 39
Base 9: 81, length 2, decimal 73, index 17
Base 10: 9001, length 4, decimal 9001, index 56
Base 11: A0000000001, length 11, decimal 259374246011, index 905
Base 12: B001, length 4, decimal 19009, index 84
Base 13: C1, length 2, decimal 157, index 31
Base 14: D01, length 3, decimal 2549, index 120
Base 15: E1, length 2, decimal 211, index 41
Base 16: F1, length 2, decimal 241, index 47
Base 17: G0001, length 5, decimal 1336337, index 3039
Base 18: H1, length 2, decimal 307, index 56
Base 19: I00000000000000000000000000001, length 30, decimal 218336795902605993201009018384568383223, index 30322
Base 20: J00000000000001, length 15, decimal 31129600000000000001, index 3160
Base 21: K1, length 2, decimal 421, index 74
Base 22: L1, length 2, decimal 463, index 82

Family y{z}: ((conjectured) exist in all bases)

Base 2: not interpretable (family should have leading zeros or trailing zeros)
Base 3: 12, length 2, decimal 5, index 1
Base 4: 23, length 2, decimal 11, index 3
Base 5: 34, length 2, decimal 19, index 5
Base 6: 45, length 2, decimal 29, index 7
Base 7: 56, length 2, decimal 41, index 9
Base 8: 6777, length 4, decimal 3583, index 55
Base 9: 78, length 2, decimal 71, index 16
Base 10: 89, length 2, decimal 89, index 20
Base 11: 9A, length 2, decimal 109, index 24
Base 12: AB, length 2, decimal 131, index 27
Base 13: BCC, length 3, decimal 2027, index 176
Base 14: CD, length 2, decimal 181, index 36
Base 15: DEEEEEEEEEEEEEE, length 15, decimal 408700964355468749, index 1252
Base 16: EF, length 2, decimal 239, index 46
Base 17: FG, length 2, decimal 271, index 51
Base 18: GHH, length 3, decimal 5507, index 178
Base 19: HIIIIII, length 7, decimal 846825857, index 17286
Base 20: IJ, length 2, decimal 379, index 67
Base 21: JK, length 2, decimal 419, index 73
Base 22: KL, length 2, decimal 461, index 81

Family 1{0}2: (not exist for bases == 0 mod 2 and bases == 1 mod 3)

Base 2: not interpretable (base 2 has no digit "2")
Base 3: 12, length 2, decimal 5, index 1
Base 4: not exist
Base 5: 12, length 2, decimal 7, index 1
Base 6: not exist
Base 7: not exist
Base 8: not exist
Base 9: 12, length 2, decimal 11, index 1
Base 10: not exist
Base 11: 12, length 2, decimal 13, index 1
Base 12: not exist
Base 13: not exist
Base 14: not exist
Base 15: 12, length 2, decimal 17, index 1
Base 16: not exist
Base 17: 12, length 2, decimal 19, index 1
Base 18: not exist
Base 19: not exist
Base 20: not exist
Base 21: 12, length 2, decimal 23, index 1
Base 22: not exist

Family 1{0}z: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 12, length 2, decimal 5, index 1
Base 4: 13, length 2, decimal 7, index 2
Base 5: 104, length 3, decimal 29, index 7
Base 6: 15, length 2, decimal 11, index 2
Base 7: 16, length 2, decimal 13, index 2
Base 8: 107, length 3, decimal 71, index 15
Base 9: 18, length 2, decimal 17, index 3
Base 10: 19, length 2, decimal 19, index 4
Base 11: 10A, length 3, decimal 131, index 26
Base 12: 1B, length 2, decimal 23, index 4
Base 13: 10C, length 3, decimal 181, index 34
Base 14: 1000000000000000D, length 17, decimal 2177953337809371149, index 606
Base 15: 1E, length 2, decimal 29, index 4
Base 16: 1F, length 2, decimal 31, index 5
Base 17: 1000G, length 5, decimal 83537, index 1348
Base 18: 100H, length 4, decimal 5849, index 185
Base 19: 1I, length 2, decimal 37, index 4
Base 20: 10J, length 3, decimal 419, index 72
Base 21: 1K, length 2, decimal 41, index 5
Base 22: 1L, length 2, decimal 43, index 6

Family {z}1: ((conjectured) exist in all bases)

Base 2: 11, length 2, decimal 3, index 1
Base 3: 21, length 2, decimal 7, index 2
Base 4: 31, length 2, decimal 13, index 4
Base 5: 44441, length 5, decimal 3121, index 20
Base 6: 51, length 2, decimal 31, index 8
Base 7: 61, length 2, decimal 43, index 10
Base 8: 7777777777771, length 13, decimal 549755813881, index 73
Base 9: 81, length 2, decimal 73, index 17
Base 10: 991, length 3, decimal 991, index 44
Base 11: AA1, length 3, decimal 1321, index 111
Base 12: BBBB1, length 5, decimal 248821, index 97
Base 13: C1, length 2, decimal 157, index 31
Base 14: DD1, length 3, decimal 2731, index 131
Base 15: E1, length 2, decimal 211, index 41
Base 16: F1, length 2, decimal 241, index 47
Base 17: GGGGGGGGGG1, length 11, decimal 34271896307617, index 8834
Base 18: H1, length 2, decimal 307, index 56
Base 19: II1, length 3, decimal 6841, index 496
Base 20: JJJJJJJJJJJJJJJJ1, length 17, decimal 13107199999999999999981, index 3185
Base 21: K1, length 2, decimal 421, index 74
Base 22: L1, length 2, decimal 463, index 82

Family {z}y: (not exist for bases == 0 mod 2)

Base 2: not interpretable (family should have leading zeros or trailing zeros)
Base 3: 21, length 2, decimal 7, index 2
Base 4: not exist
Base 5: 43, length 2, decimal 23, index 6
Base 6: not exist
Base 7: 65, length 2, decimal 47, index 11
Base 8: not exist
Base 9: 87, length 2, decimal 79, index 18
Base 10: not exist
Base 11: AAA9, length 4, decimal 14639, index 227
Base 12: not exist
Base 13: CB, length 2, decimal 167, index 33
Base 14: not exist
Base 15: ED, length 2, decimal 223, index 42
Base 16: not exist
Base 17: GGGGGF, length 6, decimal 24137567, index 4999
Base 18: not exist
Base 19: IH, length 2, decimal 359, index 64
Base 20: not exist
Base 21: KJ, length 2, decimal 439, index 77
Base 22: not exist

sweety439 2022-07-30 02:38

[QUOTE=sweety439;571906]The largest possible appearance for given digit d in minimal prime (start with b+1) in base b:

If base b has repunit primes, then the largest possible appearance for digit d=1 in minimal prime (start with b+1) in base b is the length of smallest repunit prime base b (i.e. [URL="https://oeis.org/A084740"]A084740[/URL](b)), the first bases which do not have repunit primes are 9, 25, 32, 49, 64, ...

[CODE]
b=2, d=0: 0
b=2, d=1: 2 (the prime 11)
b=3, d=0: 0
b=3, d=1: 3 (the prime 111)
b=3, d=2: 1 (the primes 12 and 21)
b=4, d=0: 0
b=4, d=1: 2 (the prime 11)
b=4, d=2: 2 (the prime 221)
b=4, d=3: 1 (the primes 13, 23, 31)
b=5, d=0: 93 (the prime 10[SUB]93[/SUB]13)
b=5, d=1: 3 (the prime 111)
b=5, d=2: 1 (the primes 12, 21, 23, 32)
b=5, d=3: 4 (the prime 33331)
b=5, d=4: 4 (the primes 14444 and 44441)
b=6, d=0: 2 (the prime 40041)
b=6, d=1: 2 (the prime 11)
b=6, d=2: 1 (the primes 21 and 25)
b=6, d=3: 1 (the primes 31 and 35)
b=6, d=4: 3 (the prime 4441)
b=6, d=5: 1 (the primes 15, 25, 35, 45, 51)
b=7, d=0: 7 (the prime 5100000001)
b=7, d=1: 5 (the prime 11111)
b=7, d=2: 3 (the prime 1222)
b=7, d=3: 16 (the prime 3[SUB]16[/SUB]1)
b=7, d=4: 2 (the primes 344, 445, 544, 4504, 40054)
b=7, d=5: 4 (the prime 35555)
b=7, d=6: 2 (the prime 6634)
b=8, d=0: 3 (the prime 500025)
b=8, d=1: 3 (the prime 111)
b=8, d=2: 2 (the prime 225)
b=8, d=3: 3 (the prime 3331)
b=8, d=4: 220 (the prime 4[SUB]220[/SUB]7)
b=8, d=5: 14 (the prime 5[SUB]13[/SUB]25)
b=8, d=6: 2 (the primes 661 and 667)
b=8, d=7: 12 (the prime 7[SUB]12[/SUB]1)
b=9, d=0: 1158 (the prime 30[SUB]1158[/SUB]11)
b=9, d=1: 36 (the prime 561[SUB]36[/SUB])
b=9, d=2: 4 (the prime 22227)
b=9, d=3: 8 (the prime 8333333335)
b=9, d=4: 11 (the prime 54[SUB]11[/SUB])
b=9, d=5: 4 (the prime 55551)
b=9, d=6: 329 (the prime 76[SUB]329[/SUB]2)
b=9, d=7: 687 (the prime 27[SUB]686[/SUB]07)
b=9, d=8: 19 (the prime 8[SUB]19[/SUB]335)
b=10, d=0: 28 (the prime 50[SUB]28[/SUB]27)
b=10, d=1: 2 (the prime 11)
b=10, d=2: 3 (the prime 2221)
b=10, d=3: 1 (the primes 13, 23, 31, 37, 43, 53, 73, 83, 349)
b=10, d=4: 2 (the prime 449)
b=10, d=5: 11 (the prime 5[SUB]11[/SUB]1)
b=10, d=6: 4 (the prime 666649)
b=10, d=7: 2 (the primes 277, 577, 727, 757, 787, 877)
b=10, d=8: 2 (the prime 881)
b=10, d=9: 3 (the prime 9949)
b=11, d=0: 126 (the prime 50[SUB]126[/SUB]57)
b=11, d=1: 17 (the prime 1[SUB]17[/SUB])
b=11, d=2: 6 (the prime 5222222)
b=11, d=3: 10 (the prime 3[SUB]10[/SUB]7)
b=11, d=4: 44 (the prime 4[SUB]44[/SUB]1)
b=11, d=5: 221 (the prime 85[SUB]220[/SUB]05]
b=11, d=6: 124 (the prime 326[SUB]124[/SUB])
b=11, d=7: 62668 (the prime 57[SUB]62668[/SUB])
b=11, d=8: 17 (the prime 8[SUB]17[/SUB]3)
b=11, d=9: 32 (the prime 9[SUB]32[/SUB]1)
b=11, d=A: 713 (the prime A[SUB]713[/SUB]58)
b=12, d=0: 39 (the prime 40[SUB]39[/SUB]77)
b=12, d=1: 2 (the prime 11)
b=12, d=2: 3 (the prime 222B)
b=12, d=3: 1 (the primes 31, 35, 37, 3B)
b=12, d=4: 3 (the prime 4441)
b=12, d=5: 2 (the primes 565 and 655)
b=12, d=6: 2 (the prime 665)
b=12, d=7: 3 (the primes 4777 and 9777)
b=12, d=8: 1 (the primes 81, 85, 87, 8B)
b=12, d=9: 4 (the prime 9999B)
b=12, d=A: 4 (the prime AAAA1)
b=12, d=B: 7 (the prime BBBBBB99B)
b=13, d=0: 32017 (the prime 80[SUB]32017[/SUB]111)
b=13, d=1: 5 (the prime 11111)
b=13, d=2: 77 (the prime 72[SUB]77[/SUB])
b=13, d=3: >82000 (the prime A3[SUB]n[/SUB]A)
b=13, d=4: 14 (the prime 94[SUB]14[/SUB])
b=13, d=5: >88000 (the prime 95[SUB]n[/SUB])
b=13, d=6: 137 (the prime 6[SUB]137[/SUB]A3)
b=13, d=7: 1504 (the prime 7[SUB]1504[/SUB]1)
b=13, d=8: 53 (the prime 8[SUB]53[/SUB]7)
b=13, d=9: 1362 (the prime 9[SUB]1362[/SUB]5)
b=13, d=A: 95 (the prime C5A[SUB]95[/SUB])
b=13, d=B: 834 (the prime B[SUB]834[/SUB]74)
b=13, d=C: 10631 (the prime C[SUB]10631[/SUB]92)
b=14, d=0: 83 (the prime 40[SUB]83[/SUB]49)
b=14, d=1: 3 (the prime 111)
b=14, d=2: 3 (the prime B2225)
b=14, d=3: 5 (the prime A33333)
b=14, d=4: 63 (the prime 4[SUB]63[/SUB]09)
b=14, d=5: 36 (the prime 85[SUB]36[/SUB])
b=14, d=6: 10 (the prime 86[SUB]10[/SUB]99)
b=14, d=7: 2 (the primes 771, 77D)
b=14, d=8: 86 (the prime 8[SUB]86[/SUB]B)
b=14, d=9: 37 (the prime 9[SUB]36[/SUB]89)
b=14, d=A: 59 (the prime A[SUB]59[/SUB]3)
b=14, d=B: 78 (the prime 6B[SUB]77[/SUB]2B)
b=14, d=C: 79 (the prime 8C[SUB]79[/SUB]3)
b=14, d=D: 19698 (the prime 4D[SUB]19698[/SUB])
b=15, d=0: 33 (the prime 50[SUB]33[/SUB]17)
b=15, d=1: 3 (the prime 111)
b=15, d=2: 9 (the prime 2222222252)
b=15, d=3: 12 (the prime 3[SUB]12[/SUB]1)
b=15, d=4: 3 (the prime 4434)
b=15, d=5: 8 (the prime 555555557)
b=15, d=6: 104 (the prime 96[SUB]104[/SUB]08)
b=15, d=7: 156 (the prime 7[SUB]155[/SUB]97)
b=15, d=8: 8 (the prime 8888888834)
b=15, d=9: 10 (the prime 9999999999D)
b=15, d=A: 4 (the prime AAAA52)
b=15, d=B: 31 (the prime EB[SUB]31[/SUB])
b=15, d=C: 10 (the prime DCCCCCCCCCC8)
b=15, d=D: 16 (the prime D[SUB]16[/SUB]B)
b=15, d=E: 145 (the prime E[SUB]145[/SUB]397)
b=16, d=0: 3542 (the prime 90[SUB]3542[/SUB]91)
b=16, d=1: 2 (the prime 11)
b=16, d=2: 32 (the prime 2[SUB]32[/SUB]7)
b=16, d=3: >76000 (the prime 3[SUB]n[/SUB]AF)
b=16, d=4: 72785 (the prime 4[SUB]72785[/SUB]DD)
b=16, d=5: 70 (the prime A015[SUB]70[/SUB])
b=16, d=6: 87 (the prime 56[SUB]87[/SUB]F)
b=16, d=7: 20 (the prime 7[SUB]19[/SUB]87)
b=16, d=8: 1517 (the prime F8[SUB]1517[/SUB]F)
b=16, d=9: 1052 (the prime D9[SUB]1052[/SUB])
b=16, d=A: 305 (the prime DA[SUB]305[/SUB]5)
b=16, d=B: 32234 (the prime DB[SUB]32234[/SUB])
b=16, d=C: 3700 (the prime 5BC[SUB]3700[/SUB]D)
b=16, d=D: 39 (the prime 4D[SUB]39[/SUB])
b=16, d=E: 34 (the prime E[SUB]34[/SUB]B)
b=16, d=F: 1961 (the prime 300F[SUB]1960[/SUB]AF)
[/CODE][/QUOTE]

The longest (not only the largest) minimal prime (start with b+1) with given first digit [I]or[/I] given last digit: (of course, first digit must not be 0, and last digit must be coprime to the base (b))

Base 2:

start with 1: 11 (length 2)
end with 1: 11 (length 2)

Base 3:

start with 1: 111 (length 3)
start with 2: 21 (length 2)
end with 1: 111 (length 3)
end with 2: 12 (length 2)

Base 4:

start with 1: 11, 13 (length 2)
start with 2: 221 (length 3)
start with 3: 31 (length 2)
end with 1: 221 (length 3)
end with 3: 13, 23 (length 2)

Base 5:

start with 1: 10[SUB]93[/SUB]13 (length 96)
start with 2: 21, 23 (length 2)
start with 3: 300031 (length 6)
start with 4: 44441 (length 5)
end with 1: 300031 (length 6)
end with 2: 12, 32 (length 2)
end with 3: 10[SUB]93[/SUB]13 (length 96)
end with 4: 14444 (length 5)

Base 6:

start with 1: 11, 15 (length 2)
start with 2: 21, 25 (length 2)
start with 3: 31, 35 (length 2)
start with 4: 40041 (length 5)
start with 5: 51 (length 2)
end with 1: 40041 (length 5)
end with 5: 15, 25, 35, 45 (length 2)

Base 7:

start with 1: 1100021 (length 7)
start with 2: 2111 (length 4)
start with 3: 33333333333333331 (length 17)
start with 4: 40054 (length 5)
start with 5: 5100000001 (length 10)
start with 6: 6034, 6634 (length 4)
end with 1: 33333333333333331 (length 17)
end with 2: 1022, 1112, 1202, 1222 (length 4)
end with 3: 300053 (length 6)
end with 4: 40054 (length 5)
end with 5: 35555 (length 5)
end with 6: 346 (length 3)

Base 8:

start with 1: 107, 111, 117, 141, 147, 161, 177 (length 3)
start with 2: 225, 255 (length 3)
start with 3: 3344441 (length 7)
start with 4: 4[SUB]220[/SUB]7 (length 221)
start with 5: 555555555555525 (length 15)
start with 6: 60171, 60411, 60741 (length 5)
start with 7: 7777777777771 (length 13)
end with 1: 7777777777771 (length 13)
end with 3: 4043, 4443 (length 4)
end with 5: 555555555555525 (length 15)
end with 7: 4[SUB]220[/SUB]7 (length 221)

Base 9:

start with 1: 1000000000000000000000000057 (length 28)
start with 2: 27[SUB]686[/SUB]07 (length 689)
start with 3: 30[SUB]1158[/SUB]11 (length 1161)
start with 4: 438 (length 3)
start with 5: 56111111111111111111111111111111111111 (length 38)
start with 6: 631111 (length 6)
start with 7: 76[SUB]329[/SUB]2 (length 331)
start with 8: 8888888888888888888335 (length 22)
end with 1: 30[SUB]1158[/SUB]11 (length 1161)
end with 2: 76[SUB]329[/SUB]2 (length 331)
end with 4: 544444444444 (length 12)
end with 5: 8888888888888888888335 (length 22)
end with 7: 27[SUB]686[/SUB]07 (length 689)
end with 8: 33388 (length 5)

Base 10:

start with 1: 11, 13, 17, 19 (length 2)
start with 2: 22000001 (length 8)
start with 3: 349 (length 3)
start with 4: 409, 449, 499 (length 3)
start with 5: 5000000000000000000000000000027 (length 31)
start with 6: 60000049, 66000049, 66600049 (length 8)
start with 7: 727, 757, 787 (length 3)
start with 8: 80555551 (length 8)
start with 9: 946669 (length 6)
end with 1: 555555555551 (length 12)
end with 3: 13, 23, 43, 53, 73, 83 (length 2)
end with 7: 5000000000000000000000000000027 (length 31)
end with 9: 60000049, 66000049, 66600049 (length 8)

Base 11:

start with 1: 10[SUB]125[/SUB]51 (length 128)
start with 2: 2888882883, 2888888883 (length 10)
start with 3: 326[SUB]122[/SUB] (length 124)
start with 4: 44777777777777777777777777777777777777777777777777777777777777777 (length 65)
start with 5: 57[SUB]62668[/SUB] (length 62669)
start with 6: 6000000000000083 (length 16)
start with 7: 7[SUB]759[/SUB]44 (length 761)
start with 8: 85[SUB]220[/SUB]05 (length 223)
start with 9: 99777777777777777777777777777777777777777777777777777777777777777 (length 65)
start with A: A[SUB]713[/SUB]58 (length 715)
end with 1: 10[SUB]125[/SUB]51 (length 128)
end with 2: 5555555555555555555555A52 (length 25)
end with 3: 5[SUB]119[/SUB]053 (length 123)
end with 4: 7[SUB]759[/SUB]44 (length 761)
end with 5: 85[SUB]220[/SUB]05 (length 223)
end with 6: 326[SUB]122[/SUB] (length 124)
end with 7: 57[SUB]62668[/SUB] (length 62669)
end with 8: A[SUB]713[/SUB]58 (length 715)
end with 9: 90000000000000000000000000000000000000009799 (length 44)
end with A: 5[SUB]161[/SUB]2A (length 163)

Base 12:

start with 1: 11, 15, 17, 1B (length 2)
start with 2: 2001, 200B, 202B, 222B, 229B, 292B, 299B (length 4)
start with 3: 31, 35, 37, 3B (length 2)
start with 4: 400000000000000000000000000000000000000077 (length 42)
start with 5: 565 (length 3)
start with 6: 600A5 (length 5)
start with 7: 7999B (length 5)
start with 8: 81, 85, 87, 8B (length 2)
start with 9: 9999B (length 5)
start with A: AA000001 (length 8)
start with B: B0000000000000000000000000009B (length 30)
end with 1: AA000001 (length 8)
end with 5: A00065 (length 6)
end with 7: 400000000000000000000000000000000000000077 (length 42)
end with B: B0000000000000000000000000009B (length 30)

sweety439 2022-07-30 02:45

1 Attachment(s)
upload data files (zipped), for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 30, 36

after unzip them, you need to rename "kernel b" to "kernel b.txt" and rename "left b" to "left b.txt", "kernel b" is the data for all known minimal primes (start with b+1) in base b, and "left b" is the data for all unsolved families in base b

search limits for the unsolved families:

base 13 family 9{5} at length 115000
base 13 family A{3}A at length 111000
base 16 family {3}AF at length 98000
all bases 17, 21, 36 families at length 20000

sweety439 2022-08-02 07:49

Just let you know, I know exactly what bases 2 <= b <= 1024 have these families unsolved: (at length 100000) (also exactly what bases 2 <= b <= 1024 have these families proven as only contain composites (only count the numbers > base (b)), by covering congruence, algebraic factorization, or combine of them)

{1}
1{0}1
2{0}1
3{0}1
4{0}1
5{0}1
6{0}1
7{0}1
8{0}1
9{0}1
A{0}1
B{0}1
C{0}1
1{z}
2{z}
3{z}
4{z}
5{z}
6{z}
7{z}
8{z}
9{z}
A{z}
B{z}
z{0}1
y{z}

Also these families, but only at length 20000:

1{0}2
1{0}3
1{0}4
{z}w
{z}x
{z}y
1{0}z
{z}1
{y}z

sweety439 2022-08-10 18:56

[QUOTE=sweety439;609890]* flexible numbers base b: Let d be a divisor (>1) of b-1 (if b-1 is prime, then d can only be b-1 itself), find the smallest prime of the form ((d-1)*b^n+1)/d with n >= 2, this prime is always minimal prime (start with b+1) base b
** for the case d = 2, such prime may not exist and it is widely believed that there are only finitely many such primes for fixed base b, since it is generalized half Fermat prime in base b
** for the case d > 2, such prime are usually expected to exist (as there cannot be covering congruence of this form, but there may be algebraic factorization or combine of covering congruence and algebraic factorization if d-1 is indeed perfect odd power (of the form m^r with odd r > 1) or of the form 4*m^4, and if d-1 is of neither of these two forms, then there must be prime of this form), but the smallest such prime may be large, e.g.

[CODE]
b,d,smallest exponent n
13,12,564
17,8,190
23,11,3762
31,6,1026
43,14,580
70,69,555
[/CODE][/QUOTE]

For the smallest flexible primes base b with given d dividing b-1:

[CODE]
4,3,2
5,4,2
6,5,2
7,3,3
7,6,2
8,7,3
9,4,2
9,8,2
10,3,2
10,9,2
11,5,2
11,10,2
12,11,2
13,3,2
13,4,2
13,6,3
13,12,564
14,13,2
15,7,2
15,14,10
16,3,3
16,5,(not exist, Aurifeuillian factorization of x^4+4*y^4)
16,15,2
17,4,9
17,8,190
17,16,2
18,17,4
19,3,2
19,6,78
19,9,13
19,18,14
20,19,2
21,4,2
21,5,2
21,10,2
21,20,2
22,3,6
22,7,3
22,21,2
23,11,3762
23,22,8
24,23,4
25,3,4
25,4,3
25,6,2
25,8,2
25,12,3
25,24,2
26,5,2
26,25,5
27,13,2
27,26,2
28,3,2
28,9,4
28,27,3
29,4,2
29,7,4
29,14,6
29,28,2
30,29,6
31,3,2
31,5,2
31,6,1026
31,10,24
31,15,99
31,30,2
32,31,2
33,4,3
33,8,2
33,16,2
33,32,252
34,3,3
34,11,2
34,33,3
35,17,2
35,34,20
36,5,45
36,7,5
36,35,2
37,3,3
37,4,6
37,6,4
37,9,2
37,12,4
37,18,12
37,36,6
38,37,4
39,19,3
39,38,2
40,3,3
40,13,3
40,39,2
41,4,3
41,5,6
41,8,2
41,10,15
41,20,2
41,40,4
42,41,2
43,3,12
43,6,38
43,7,4
43,14,580
43,21,3
43,42,24
44,43,3
45,4,28
45,11,5
45,22,2
45,44,2
46,3,3
46,5,2
46,9,7
46,15,3
46,45,2
47,23,2
47,46,2
48,47,4
49,3,2
49,4,2
49,6,3
49,8,8
49,12,26
49,16,2
49,24,4
49,48,2
50,7,2
50,49,3
51,5,2
51,10,2
51,25,5
51,50,2
52,3,3
52,17,5
52,51,17
53,4,4
53,13,2
53,26,4
53,52,24
54,53,2
55,3,2
55,6,2
55,9,2
55,18,2
55,27,6
55,54,2
56,5,78
56,11,2
56,55,2
57,4,2
57,7,5
57,8,2
57,14,6
57,28,44
57,56,2
58,3,2
58,19,2
58,57,3
59,29,2
59,58,4
60,59,2
61,3,10
61,4,2
61,5,6
61,6,3
61,10,4
61,12,6
61,15,11
61,20,70
61,30,4
61,60,2
62,61,4
63,31,3
63,62,4
64,3,2
64,7,2
64,9,(not exist, sum-of-two-cubes factorization)
64,21,24
64,63,11
65,4,2
65,8,2
65,16,5
65,32,2
65,64,2
66,5,15
66,13,2
66,65,2
67,3,6
67,6,6
67,11,19
67,22,(>10000)
67,33,3
67,66,2
68,67,5
69,4,2
69,17,2
69,34,2
69,68,2
70,3,4
70,23,11
70,69,555
71,5,22
71,7,3
71,10,836
71,14,14
71,35,6
71,70,2
72,71,5
73,3,4
73,4,4
73,6,2
73,8,2
73,9,28
73,12,4
73,18,9
73,24,2
73,36,85
73,72,8
74,73,10
75,37,24
75,74,12
76,3,2
76,5,2
76,15,3
76,25,3
76,75,3
77,4,2
77,19,6
77,38,4
77,76,2
78,7,8
78,11,2
78,77,3
79,3,61
79,6,162
79,13,4
79,26,8
79,39,213
79,78,6
80,79,24
81,4,3
81,5,(not exist, Aurifeuillian factorization of x^4+4*y^4)
81,8,2
81,10,29
81,16,2
81,20,5
81,40,2
81,80,4
82,3,2
82,9,10
82,27,3
82,81,6
83,41,104
83,82,680
84,83,2
85,3,2
85,4,2
85,6,13
85,7,4
85,12,6
85,14,2
85,21,57
85,28,2
85,42,10
85,84,6
86,5,6
86,17,2
86,85,2
87,43,2
87,86,2
88,3,28
88,29,2
88,87,3
89,4,6
89,8,6
89,11,288
89,22,2
89,44,2
89,88,132
90,89,2
91,3,2
91,5,9
91,6,4
91,9,5
91,10,36
91,15,4
91,18,4
91,30,8
91,45,16
91,90,140
92,7,51
92,13,4
92,91,4
93,4,156
93,23,2
93,46,2
93,92,4
94,3,51
94,31,12
94,93,2
95,47,8
95,94,2
96,5,3
96,19,2
96,95,84
97,3,7
97,4,2
97,6,2
97,8,2
97,12,???
97,16,2
97,24,9
97,32,4
97,48,28
97,96,2
98,97,137
99,7,6
99,14,6
99,49,2
99,98,6
100,3,3
100,9,7
100,11,2
100,33,2
100,99,5
[/CODE]

sweety439 2022-08-13 13:13

5 Attachment(s)
Results for bases b>16

Base 17 is searched to length 32100
Base 18 is proven (including the primality of the primes)
Base 19 is searched to length 20000

sweety439 2022-08-13 13:16

5 Attachment(s)
Base 20 is proven (including the primality of the primes)
Base 21 is searched to length 20000
Base 22 is proven except the primality of the large strong probable prime B(K^22001)5
Base 23 is now reserved
Base 24 is proven (including the primality of the primes)

sweety439 2022-08-13 13:19

3 Attachment(s)
Base 30 is proven except the primality of the large strong probable prime I(0^24608)D
Base 36 is searched to length 20000

sweety439 2022-08-13 17:17

The base 19 unsolved family 5{H}5 is very low [URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]) but eventually should yield a prime (see [URL="http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show"]http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show[/URL])

Its formula is (107*19^(n+1)-233)/18

(since neither 107 nor 233 is perfect power ([URL="https://oeis.org/A001597"]https://oeis.org/A001597[/URL]), this family has no algebraic factors)

* n == 0 mod 2: factor of 2 (also factor of 4, 5, 10, 20)

this only left n == 1 mod 2

* n == 1 mod 3: factor of 3

this only left n == 3, 5 mod 6

* n == 5 mod 6: factor of 7

this only left n == 3 mod 6

* n == 9 mod 12: factor of 13

this only left n == 3 mod 12

* n == 3 mod 8: factor of 17

this only left n == 15 mod 24

the n = 15 number is divisible by many small primes (11, 29, 47, 71), but all of these primes (p) have large and non-smooth order (znorder(Mod(19,p))), the n = 39 number is divisible by 281, and the n = 63 and 87 numbers have no small prime factors, the n = 111 number is divisible by 89

Base 26 is fully searched to length 20000, and there are 25250 known minimal (probable) primes (start with b+1) and 9 unsolved families, and base 28 is [URL="https://mersenneforum.org/showpost.php?p=546078&postcount=3659"]technically[/URL] fully searched to length 543203 (if we allow probable primes in place of proven primes, see [URL="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf"]http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf[/URL] (see section 3: Recent Results Establishing the Mixed Sierpinski Theorem) and [URL="https://oeis.org/history?seq=A004023&start=50"]https://oeis.org/history?seq=A004023&start=50[/URL] (see M. F. Hasler's discussion in the pink box)), and there are 25528 known minimal (probable) primes (start with b+1), and the only one unsolved family is O{A}F (see [URL="https://github.com/xayahrainie4793/quasi-mepn-data"]https://github.com/xayahrainie4793/quasi-mepn-data[/URL] and [URL="https://github.com/curtisbright/mepn-data/blob/master/data/sieve.28.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/sieve.28.txt[/URL]), interestingly, in base 28 there are only 3 known minimal primes (start with b+1) (and it is likely totally 4 minimal primes (start with b+1)) with length > 5271: N(6^24051)LR, 5O(A^31238)F, O4(O^94535)9, and more interestingly, base 28 seems to be high weight base, but there are many families (which [I]must[/I] be minimal primes (start with b+1) in all bases b, if these families are interpretable in this base b) whose smallest length (to make the number prime) (see [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]) set records: 3*b^n+1 (3{0}1, length 8, index 19858), b^n-3 ({z}x, which is {R}P in base 28, length 10, index 23827), b^n-5 ({z}v, which is {R}N in base 28, length 60, index 25401), (b/2)*b^n-1 (#{z}, which is D{R} in base 28, length 48, index 25367), {2}1 (length 40, index 25337)

sweety439 2022-08-13 20:22

Integers b>=2 sorted by [URL="https://oeis.org/A062955"]A062955[/URL](b):

2 (1), 3 (4), 4 (6), 6 (10), 5 (16), 8 (28), 7&10 (36), 12 (44), 9 (48), 14 (78), 11 (100), 18 (102), 15 (112), 16 (120), 13 (144), 20 (152), 24 (184), 22 (210), 30 (232), 21 (240), 17 (256), 26 (300), 19&28 (324), 36 (420), 27 (468), 25 (480), 23 (484), 42 (492), 32 (496), 34 (528), 40 (624), 33 (640), 38 (666), 48 (752), 29 (784), 35 (816), 44 (860), 31 (900), 39 (912), 60 (944), 54 (954), 50 (980), 46 (990), ...

Integers b>=2 sorted by number of minimal primes (starting with b+1) base b: (not sure if 26 and 28 are before 17 and 21)

2 (1), 3 (3), 4 (5), 6 (11), 5 (22), 7 (71), 8 (75), 10 (77), 12 (106), 9 (151), 18 (549), 14 (650), 11 (1068), 15 (1284), 16 (2346~2347), 30 (2619), 13 (3195~3197), 20 (3314), 24 (3409), 22 (8003), 17 (10405~10428), 21 (13373~13395), ...

Integers b>=2 sorted by length of largest minimal prime (starting with b+1) base b:

2 (2), 3&4 (3), 6 (5), 7 (17), 10 (31), 12 (42), 5 (96), 15 (157), 8 (221), 9 (1161), 18&20 (6271), 24 (8134), 14 (19699), 22 (22003), 30 (34206), 11 (62669), ...

These three sequences are conjectured to be similar, the integers b = 7 and b = 15 for the third sequence is relatively small since they (as well as b = 3) are high-weight bases (like [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] bases R7, R15, S7, S15, they are high-weight bases), i.e. they are very "primeful", while b = 5 and b = 8 and b = 11 and b = 14 are relatively large, since they are low-weight bases (like [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL], bases == 2 mod 3 are low-weight bases), although this does not hold for b = 20, which is also == 2 mod 3

sweety439 2022-08-13 20:35

[QUOTE=sweety439;609695]Number of totally digits of minimal primes (start with b+1) in base b

Sum of all minimal primes (start with b+1) in base b

Product of all minimal primes (start with b+1) in base b

Base 2:

1 primes, totally 2 digits, [URL="http://factordb.com/index.php?id=3"]sum[/URL], [URL="http://factordb.com/index.php?id=3"]product[/URL]

Base 3:

3 primes, totally 7 digits, [URL="http://factordb.com/index.php?id=25"]sum[/URL], [URL="http://factordb.com/index.php?id=455"]product[/URL]

Base 4:

5 primes, totally 11 digits, [URL="http://factordb.com/index.php?id=77"]sum[/URL], [URL="http://factordb.com/index.php?id=205205"]product[/URL]

Base 5:

22 primes, totally 169 digits, [URL="http://factordb.com/index.php?id=1100000003799642708"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457822814"]product[/URL]

Base 6:

11 primes, totally 29 digits, [URL="http://factordb.com/index.php?id=7401"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457821560"]product[/URL]

Base 7:

71 primes, totally 288 digits, [URL="http://factordb.com/index.php?id=116315467894207"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457825324"]product[/URL]

Base 8:

75 primes, totally 523 digits, [URL="http://factordb.com/index.php?id=1100000003799644593"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002371473795"]product[/URL]

Base 9:

151 primes, totally 3004 digits, [URL="http://factordb.com/index.php?id=1100000003799645271"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003450366253"]product[/URL]

Base 10:

77 primes, totally 310 digits, [URL="http://factordb.com/index.php?id=1100000003799645582"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002370859491"]product[/URL]

Base 11:

1068 primes, totally 75414 digits, [URL="http://factordb.com/index.php?id=1100000003799646641"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003583737715"]product[/URL]

Base 12:

106 primes, totally 433 digits, [URL="http://factordb.com/index.php?id=1100000003799647067"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000002457818232"]product[/URL]

Base 14:

650 primes, totally 25404 digits, [URL="http://factordb.com/index.php?id=1100000003799647609"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003575953976"]product[/URL]

Base 15:

1284 primes, totally 8286 digits, [URL="http://factordb.com/index.php?id=1100000003799647942"]sum[/URL], [URL="http://factordb.com/index.php?id=1100000003588261354"]product[/URL]

Base 18:

Conjecture: the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers.[/QUOTE]

Some interesting sequences: (since I have a conjecture that the sum of all minimal primes (start with b+1) base b is always in [URL="https://oeis.org/A063538"]https://oeis.org/A063538[/URL], i.e. it must have a prime factor >= its square root, I have run the "greatest prime factor ^2-1" sequences for them, while it is meaningless for running Aliquot sequences and home prime sequences for them)

Base 5:

[URL="http://factordb.com/sequences.php?se=1&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=5&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=15&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=2679246027472911769510990558973105766564587654898362954650162758626573330147163259079412210227479020387664869881&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=2524354896707237777317531408904915934954260592348873615264892600018&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

Base 7:

[URL="http://factordb.com/sequences.php?se=1&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=7&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=17&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=52097983347885996929656234223871153222827665000736129336122654603168136960994681369054172651120331932480926136169970911756631785631053694082780811039372170805490290148867089427484357623139878142463964718964496055675169&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=116315467894207&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

Base 8:

[URL="http://factordb.com/sequences.php?se=1&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=8&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=18&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=601422512652031577041731644690688085382970325466673117063892293282377170111033197757433598116289647462255153225617550228771252606639501695222722413777622007424501602330972216951740697427538717237667339336079556094694792181591620097055442212841704601636625618211036725618077934080240342815118003410486041788181848738839465698630637694531177622714963004039096694901589679999544722946834140279663847002903753153074508353867176240552187410351876749357222807&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064067632208136437&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence start with sum[/URL]

Base 10:

[URL="http://factordb.com/sequences.php?se=1&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=10&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=20&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]inverse home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=908782245413077872778349332865297541016467712261263904090540925527772489077630963754620648417435095466161914246058996597429566922146944042476947622265170645254580275156275420407956842180346477122878050807047526191345015641497667736008418546839127294348428931336330299438242385794206911671&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=5000000000000000000555857895791&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence start with sum[/URL]

Base 12: (no inverse home prime sequence available)

[URL="http://factordb.com/sequences.php?se=1&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]aliquot sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=29&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]home prime sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=295379858346459160321805517996534469599028406080898251334958136688449432694413591686670702771988538717608698226463191652828824496701353686057825770706609766177376504079608957349577470384478406844265668777103440858249254803548508509588031749392867767312098917190750874886680388900285905326905017101113635039084401476984289722816777990643065656223617398811312978513004322459271352884623281096212482243698557359065144801241770738235059141453&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with product[/URL] [URL="http://factordb.com/sequences.php?se=24&aq=705490352625379091234039122063021997484453506&action=last20&fr=0&to=100"]greatest prime factor ^2-1 sequence starting with sum[/URL]

sweety439 2022-08-13 20:40

For the interest of "greatest prime factor ^2-1" sequences:

* [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality proving[/URL]
* [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality proving[/URL]
* [URL="https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm"]P-1 integer factorization method[/URL]
* [URL="https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm"]P+1 integer factorization method[/URL]

[URL="https://oeis.org/A087713"]https://oeis.org/A087713[/URL] (greatest prime factor of p^2-1)
[URL="https://oeis.org/A024710"]https://oeis.org/A024710[/URL] (the same sequence (start with p=11) of the A087713, which is the greatest prime factor of A024702)
[URL="https://oeis.org/A024702"]https://oeis.org/A024702[/URL] ((p^2-1)/24)
[URL="https://oeis.org/A084920"]https://oeis.org/A084920[/URL] (p^2-1)
[URL="https://oeis.org/A001248"]https://oeis.org/A001248[/URL] (p^2)
[URL="https://oeis.org/A001318"]https://oeis.org/A001318[/URL] (generalized pentagonal numbers, (n^2-1)/24)


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