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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 2020-12-12 10:56

* Case (5,1):

** [B]51[/B] is prime, and thus the only minimal prime in this family.

* Case (5,3):

** [B]53[/B] is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)

*** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)

**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

**** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)

***** 500525 is not prime.

***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B]

***** The smallest prime of the form 5{5}025 is [B]55555025[/B]

***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025 (not minimal prime, since 55555025 and 555555555555525 are primes)

***** The smallest prime of the form 5{5}0525 is [B]5550525[/B]

***** The smallest prime of the form 5{5}00525 is [B]5500525[/B]

***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)

* Case (5,7):

** [B]57[/B] is prime, and thus the only minimal prime in this family.

sweety439 2020-12-13 05:06

* Case (6,1):

** Since 65, 21, 51, [B]631[/B], [B]661[/B] are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Since 111, 141, 401, 471, 701, 711, [B]6101[/B], [B]6441[/B] are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes)

**** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes)

***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, [B]60411[/B] are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes)

****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, [B]60741[/B] are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes)

***** Since 471, [B]60171[/B] is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes)

****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime.

**** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11

***** For the 6{0}1{7}1 family, since [B]60171[/B] is prime, and thus the only minimal prime in the family 6{0}1{7}1.

***** For the 6{0,4}11 family, since 401, 6441, [B]60411[/B] are primes, we only need to consider the number 6411 and the family 6{0}11

****** 6411 is not prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

**** For the 6{0,7}4{1}1 family, since [B]60411[/B] is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41

***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411

****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes.

***** For the 6{0,7}41 family, since 701, 6777, [B]60741[/B] are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes)

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

****** Neither of 6741, 67741 are primes.

***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes)

****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes)

******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71

******** Since 717 and [B]60171[/B] are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes)

********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes)

********** 6171 is not prime.

****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes.

sweety439 2020-12-13 05:38

1 Attachment(s)
Upload past file, the set is not complete for bases >=7, I want to complete them.

I know some primes in the set which is not listed:

* base 7: 33333333333333331

* base 8: 77774444441, 7777777777771, 555555555555525, 4[SUB]220[/SUB]7

* base 10: 555555555551

* base 11: A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 (they are in the searching where single-digit primes are included, but this puzzle does not include single-digit primes)

* base 13: 80[SUB]32017[/SUB]111

* base 14: 4D[SUB]19698[/SUB]

* base 15: DE[SUB]14[/SUB]

* base 17: 74[SUB]4904[/SUB]

(this problem is much harder than the original minimal prime (where single-digit primes are included), see post [URL="https://mersenneforum.org/showpost.php?p=564315&postcount=57"]https://mersenneforum.org/showpost.php?p=564315&postcount=57[/URL])

sweety439 2020-12-13 07:53

Found a minimal prime (start with 2 digits) in base 13: 7[SUB]1504[/SUB]1, which equals (7*13^1505-79)/12

sweety439 2020-12-13 09:46

1 Attachment(s)
Consider the "simplest" families x{y} and {x}y, where x,y are base b digits

Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1

[CODE]
b, x, y, smallest prime
2, {1}, 1: 3
2, 1, {1}: 3
3, {1}, 1: 13
3, 1, {1}: 13
3, {1}, 2: 5
3, 1, {2}: 5
3, {2}, 1: 7
3, 2, {1}: 7
4, {1}, 1: 5
4, 1, {1}: 5
4, {1}, 3: 7
4, 1, {3}: 7
4, {2}, 1: 41
4, 2, {1}: 37
4, {2}, 3: 11
4, 2, {3}: 11
4, {3}, 1: 13
4, 3, {1}: 13
5, {1}, 1: 31
5, 1, {1}: 31
5, {1}, 2: 7
5, 1, {2}: 7
5, {1}, 3: 0
5, 1, {3}: 43
5, {1}, 4: 0
5, 1, {4}: 1249
5, {2}, 1: 11
5, 2, {1}: 11
5, {2}, 3: 13
5, 2, {3}: 13
5, {3}, 1: 2341
5, 3, {1}: 0
5, {3}, 2: 17
5, 3, {2}: 17
5, {3}, 4: 19
5, 3, {4}: 19
5, {4}, 1: 3121
5, 4, {1}: 0
5, {4}, 3: 23
5, 4, {3}: 23
6, {1}, 1: 7
6, 1, {1}: 7
6, {1}, 5: 11
6, 1, {5}: 11
6, {2}, 1: 13
6, 2, {1}: 13
6, {2}, 5: 17
6, 2, {5}: 17
6, {3}, 1: 19
6, 3, {1}: 19
6, {3}, 5: 23
6, 3, {5}: 23
6, {4}, 1: 1033
6, 4, {1}: 151
6, {4}, 5: 29
6, 4, {5}: 29
6, {5}, 1: 31
6, 5, {1}: 31
7, {1}, 1: 2801
7, 1, {1}: 2801
7, {1}, 2: 401
7, 1, {2}: 457
7, {1}, 3: 59
7, 1, {3}: 73
7, {1}, 4: 11
7, 1, {4}: 11
7, {1}, 5: 61
7, 1, {5}: 89
7, {1}, 6: 13
7, 1, {6}: 13
7, {2}, 1: 113
7, 2, {1}: 743
7, {2}, 3: 17
7, 2, {3}: 17
7, {2}, 5: 19
7, 2, {5}: 19
7, {3}, 1: 116315256993601
7, 3, {1}: 7603
7, {3}, 2: 23
7, 3, {2}: 23
7, {3}, 4: 1201
7, 3, {4}: 179
7, {3}, 5: 173
7, 3, {5}: 9203
7, {4}, 1: 29
7, 4, {1}: 29
7, {4}, 3: 31
7, 4, {3}: 31
7, {4}, 5: 229
7, 4, {5}: 1657
7, {5}, 1: 281
7, 5, {1}: 8413470255870653
7, {5}, 2: 37
7, 5, {2}: 37
7, {5}, 3: 283
7, 5, {3}: 269
7, {5}, 4: 1999
7, 5, {4}: 277
7, {5}, 6: 41
7, 5, {6}: 41
7, {6}, 1: 43
7, 6, {1}: 43
7, {6}, 5: 47
7, 6, {5}: 47
8, {1}, 1: 73
8, 1, {1}: 73
8, {1}, 3: 11
8, 1, {3}: 11
8, {1}, 5: 13
8, 1, {5}: 13
8, {1}, 7: 79
8, 1, {7}: 127
8, {2}, 1: 17
8, 2, {1}: 17
8, {2}, 3: 19
8, 2, {3}: 19
8, {2}, 5: 149
8, 2, {5}: 173
8, {2}, 7: 23
8, 2, {7}: 23
8, {3}, 1: 1753
8, 3, {1}: 1609
8, {3}, 5: 29
8, 3, {5}: 29
8, {3}, 7: 31
8, 3, {7}: 31
8, {4}, 1: 76695841
8, 4, {1}: 284694975049
8, {4}, 3: 2339
8, 4, {3}: 283
8, {4}, 5: 37
8, 4, {5}: 37
8, {4}, 7: 21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064041941947533607
8, 4, {7}: 20479
8, {5}, 1: 41
8, 5, {1}: 41
8, {5}, 3: 43
8, 5, {3}: 43
8, {5}, 7: 47
8, 5, {7}: 47
8, {6}, 1: 433
8, 6, {1}: 56657856797822194249
8, {6}, 5: 53
8, 6, {5}: 53
8, {6}, 7: 439
8, 6, {7}: 3583
8, {7}, 1: 549755813881
8, 7, {1}: 457
8, {7}, 3: 59
8, 7, {3}: 59
8, {7}, 5: 61
8, 7, {5}: 61
9, {1}, 1: 0
9, 1, {1}: 0
9, {1}, 2: 11
9, 1, {2}: 11
9, {1}, 4: 13
9, 1, {4}: 13
9, {1}, 5: 0
9, 1, {5}: 131
9, {1}, 7: 97
9, 1, {7}: 151
9, {1}, 8: 17
9, 1, {8}: 17
9, {2}, 1: 19
9, 2, {1}: 19
9, {2}, 5: 23
9, 2, {5}: 23
9, {2}, 7: 14767
9, 2, {7}: 0
9, {3}, 1: 271
9, 3, {1}: 0
9, {3}, 2: 29
9, 3, {2}: 29
9, {3}, 4: 31
9, 3, {4}: 31
9, {3}, 5: 0
9, 3, {5}: 293
9, {3}, 7: 277
9, 3, {7}: 313
9, {3}, 8: 0
9, 3, {8}: 0
9, {4}, 1: 37
9, 4, {1}: 37
9, {4}, 5: 41
9, 4, {5}: 41
9, {4}, 7: 43
9, 4, {7}: 43
9, {5}, 1: 36901
9, 5, {1}: 0
9, {5}, 2: 47
9, 5, {2}: 47
9, {5}, 4: 4099
9, 5, {4}: 172595827849
9, {5}, 7: 457
9, 5, {7}: 0
9, {5}, 8: 53
9, 5, {8}: 53
9, {6}, 1: 541
9, 6, {1}: 0
9, {6}, 5: 59
9, 6, {5}: 59
9, {6}, 7: 61
9, 6, {7}: 61
9, {7}, 1: 631
9, 7, {1}: 577
9, {7}, 2: 0
9, 7, {2}: 587
9, {7}, 4: 67
9, 7, {4}: 67
9, {7}, 5: 0
9, 7, {5}: 617
9, {7}, 8: 71
9, 7, {8}: 71
9, {8}, 1: 73
9, 8, {1}: 73
9, {8}, 5: 0
9, 8, {5}: 6287
9, {8}, 7: 79
9, 8, {7}: 79
10, {1}, 1: 11
10, 1, {1}: 11
10, {1}, 3: 13
10, 1, {3}: 13
10, {1}, 7: 17
10, 1, {7}: 17
10, {1}, 9: 19
10, 1, {9}: 19
10, {2}, 1: 2221
10, 2, {1}: 211
10, {2}, 3: 23
10, 2, {3}: 23
10, {2}, 7: 227
10, 2, {7}: 277
10, {2}, 9: 29
10, 2, {9}: 29
10, {3}, 1: 31
10, 3, {1}: 31
10, {3}, 7: 37
10, 3, {7}: 37
10, {4}, 1: 41
10, 4, {1}: 41
10, {4}, 3: 43
10, 4, {3}: 43
10, {4}, 7: 47
10, 4, {7}: 47
10, {4}, 9: 449
10, 4, {9}: 499
10, {5}, 1: 555555555551
10, 5, {1}: 511111
10, {5}, 3: 53
10, 5, {3}: 53
10, {5}, 7: 557
10, 5, {7}: 577
10, {5}, 9: 59
10, 5, {9}: 59
10, {6}, 1: 61
10, 6, {1}: 61
10, {6}, 7: 67
10, 6, {7}: 67
10, {7}, 1: 71
10, 7, {1}: 71
10, {7}, 3: 73
10, 7, {3}: 73
10, {7}, 9: 79
10, 7, {9}: 79
10, {8}, 1: 881
10, 8, {1}: 811
10, {8}, 3: 83
10, 8, {3}: 83
10, {8}, 7: 887
10, 8, {7}: 877
10, {8}, 9: 89
10, 8, {9}: 89
10, {9}, 1: 991
10, 9, {1}: 911
10, {9}, 7: 97
10, 9, {7}: 97
11, {1}, 1: 50544702849929377
11, 1, {1}: 50544702849929377
11, {1}, 2: 13
11, 1, {2}: 13
11, {1}, 3: 0
11, 1, {3}: 157
11, {1}, 4: 0
11, 1, {4}: 7783884238889124073
11, {1}, 5: 137
11, 1, {5}: 181
11, {1}, 6: 17
11, 1, {6}: 17
11, {1}, 7: 139
11, 1, {7}: 24889
11, {1}, 8: 19
11, 1, {8}: 19
11, {1}, 9: 0
11, 1, {9}: 229
11, {1}, 10: 0
11, 1, {10}: 241
11, {2}, 1: 23
11, 2, {1}: 23
11, {2}, 3: 354313
11, 2, {3}: 3061
11, {2}, 5: 269
11, 2, {5}: 0
11, {2}, 7: 29
11, 2, {7}: 29
11, {2}, 9: 31
11, 2, {9}: 31
11, {3}, 1: 397
11, 3, {1}: 0
11, {3}, 2: 4391
11, 3, {2}: 4259
11, {3}, 4: 37
11, 3, {4}: 37
11, {3}, 5: 401
11, 3, {5}: 0
11, {3}, 7: 85593501187
11, 3, {7}: 0
11, {3}, 8: 41
11, 3, {8}: 41
11, {3}, 10: 43
11, 3, {10}: 43
11, {4}, 1: 29156193474041220857161146715104735751776055777
11, 4, {1}: 0
11, {4}, 3: 47
11, 4, {3}: 47
11, {4}, 5: 5857
11, 4, {5}: 724729
11, {4}, 7: 114124668247
11, 4, {7}: 0
11, {4}, 9: 53
11, 4, {9}: 53
11, {5}, 1: 661
11, 5, {1}: 617
11, {5}, 2: 0
11, 5, {2}: 9212117
11, {5}, 3: 0
11, 5, {3}: 641
11, {5}, 4: 59
11, 5, {4}: 59
11, {5}, 6: 61
11, 5, {6}: 61
11, {5}, 7: 80527
11, 5, {7}: 0
11, {5}, 8: 0
11, 5, {8}: 701
11, {5}, 9: 0
11, 5, {9}: 86381
11, {6}, 1: 67
11, 6, {1}: 67
11, {6}, 5: 71
11, 6, {5}: 71
11, {6}, 7: 73
11, 6, {7}: 73
11, {7}, 1: 42811363313890182397
11, 7, {1}: 859
11, {7}, 2: 79
11, 7, {2}: 79
11, {7}, 3: 0
11, 7, {3}: 883
11, {7}, 4: 0
11, 7, {4}: 108343
11, {7}, 5: 929
11, 7, {5}: 907
11, {7}, 6: 83
11, 7, {6}: 83
11, {7}, 8: 150051217
11, 7, {8}: 114199
11, {7}, 9: 0
11, 7, {9}: 115663
11, {7}, 10: 0
11, 7, {10}: 967
11, {8}, 1: 89
11, 8, {1}: 89
11, {8}, 3: 4447933850793785179
11, 8, {3}: 11047
11, {8}, 5: 1061
11, 8, {5}: 0
11, {8}, 7: 1063
11, 8, {7}: 11579
11, {8}, 9: 97
11, 8, {9}: 97
11, {9}, 1: 20902638977899027326901591016678209
11, 9, {1}: 0
11, {9}, 2: 101
11, 9, {2}: 101
11, {9}, 4: 103
11, 9, {4}: 103
11, {9}, 5: 1193
11, 9, {5}: 0
11, {9}, 7: 2122152919
11, 9, {7}: 0
11, {9}, 8: 107
11, 9, {8}: 107
11, {9}, 10: 109
11, 9, {10}: 109
11, {10}, 1: 1321
11, 10, {1}: 0
11, {10}, 3: 113
11, 10, {3}: 113
11, {10}, 7: 1327
11, 10, {7}: 0
11, {10}, 9: 14639
11, 10, {9}: 80662724392413945103199
12, {1}, 1: 13
12, 1, {1}: 13
12, {1}, 5: 17
12, 1, {5}: 17
12, {1}, 7: 19
12, 1, {7}: 19
12, {1}, 11: 23
12, 1, {11}: 23
12, {2}, 1: 313
12, 2, {1}: 3613
12, {2}, 5: 29
12, 2, {5}: 29
12, {2}, 7: 31
12, 2, {7}: 31
12, {2}, 11: 3779
12, 2, {11}: 431
12, {3}, 1: 37
12, 3, {1}: 37
12, {3}, 5: 41
12, 3, {5}: 41
12, {3}, 7: 43
12, 3, {7}: 43
12, {3}, 11: 47
12, 3, {11}: 47
12, {4}, 1: 7537
12, 4, {1}: 7069
12, {4}, 5: 53
12, 4, {5}: 53
12, {4}, 7: 631
12, 4, {7}: 8011
12, {4}, 11: 59
12, 4, {11}: 59
12, {5}, 1: 61
12, 5, {1}: 61
12, {5}, 7: 67
12, 5, {7}: 67
12, {5}, 11: 71
12, 5, {11}: 71
12, {6}, 1: 73
12, 6, {1}: 73
12, {6}, 5: 941
12, 6, {5}: 929
12, {6}, 7: 79
12, 6, {7}: 79
12, {6}, 11: 83
12, 6, {11}: 83
12, {7}, 1: 1093
12, 7, {1}: 1021
12, {7}, 5: 89
12, 7, {5}: 89
12, {7}, 11: 1103
12, 7, {11}: 1151
12, {8}, 1: 97
12, 8, {1}: 97
12, {8}, 5: 101
12, 8, {5}: 101
12, {8}, 7: 103
12, 8, {7}: 103
12, {8}, 11: 107
12, 8, {11}: 107
12, {9}, 1: 109
12, 9, {1}: 109
12, {9}, 5: 113
12, 9, {5}: 113
12, {9}, 7: 16963
12, 9, {7}: 16651
12, {9}, 11: 203591
12, 9, {11}: 1439
12, {10}, 1: 226201
12, 10, {1}: 1453
12, {10}, 7: 127
12, 10, {7}: 127
12, {10}, 11: 131
12, 10, {11}: 131
12, {11}, 1: 248821
12, 11, {1}: 1597
12, {11}, 5: 137
12, 11, {5}: 137
12, {11}, 7: 139
12, 11, {7}: 139
13, {1}, 1: 30941
13, 1, {1}: 30941
13, {1}, 2: 2381
13, 1, {2}: 197
13, {1}, 3: 883708283
13, 1, {3}: 211
13, {1}, 4: 17
13, 1, {4}: 17
13, {1}, 5: 0
13, 1, {5}: 239
13, {1}, 6: 19
13, 1, {6}: 19
13, {1}, 7: 0
13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561
13, {1}, 8: 0
13, 1, {8}: 281
13, {1}, 9: 191
13, 1, {9}: 27130132404659193376721686434661
13, {1}, 10: 23
13, 1, {10}: 23
13, {1}, 11: 193
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17, 16, {7}: 0
17, {16}, 9: 281
17, 16, {9}: 281
17, {16}, 11: 283
17, 16, {11}: 283
17, {16}, 13: 4909
17, 16, {13}: 0
17, {16}, 15: 24137567
17, 16, {15}: 66886068539071498820247358361862720864806052666582265636907882027208271253
18, {1}, 1: 19
18, 1, {1}: 19
18, {1}, 5: 23
18, 1, {5}: 23
18, {1}, 7: 349
18, 1, {7}: 457
18, {1}, 11: 29
18, 1, {11}: 29
18, {1}, 13: 31
18, 1, {13}: 31
18, {1}, 17: 359
18, 1, {17}: 647
18, {2}, 1: 37
18, 2, {1}: 37
18, {2}, 5: 41
18, 2, {5}: 41
18, {2}, 7: 43
18, 2, {7}: 43
18, {2}, 11: 47
18, 2, {11}: 47
18, {2}, 13: 72025897
18, 2, {13}: 3198298525119427
18, {2}, 17: 53
18, 2, {17}: 53
18, {3}, 1: 18523
18, 3, {1}: 991
18, {3}, 5: 59
18, 3, {5}: 59
18, {3}, 7: 61
18, 3, {7}: 61
18, {3}, 11: 108038837
18, 3, {11}: 1181
18, {3}, 13: 67
18, 3, {13}: 67
18, {3}, 17: 71
18, 3, {17}: 71
18, {4}, 1: 73
18, 4, {1}: 73
18, {4}, 5: 1373
18, 4, {5}: 47321007179
18, {4}, 7: 79
18, 4, {7}: 79
18, {4}, 11: 83
18, 4, {11}: 83
18, {4}, 13: 1381
18, 4, {13}: 1543
18, {4}, 17: 89
18, 4, {17}: 89
18, {5}, 1: 30871
18, 5, {1}: 34130064295121260303
18, {5}, 7: 97
18, 5, {7}: 97
18, {5}, 11: 101
18, 5, {11}: 101
18, {5}, 13: 103
18, 5, {13}: 103
18, {5}, 17: 107
18, 5, {17}: 107
18, {6}, 1: 109
18, 6, {1}: 109
18, {6}, 5: 113
18, 6, {5}: 113
18, {6}, 7: 3889397851
18, 6, {7}: 412073923449193
18, {6}, 11: 2063
18, 6, {11}: 2153
18, {6}, 13: 37057
18, 6, {13}: 39451
18, {6}, 17: 2069
18, 6, {17}: 2267
18, {7}, 1: 127
18, 7, {1}: 127
18, {7}, 5: 131
18, 7, {5}: 131
18, {7}, 11: 137
18, 7, {11}: 137
18, {7}, 13: 139
18, 7, {13}: 139
18, {7}, 17: 2411
18, 7, {17}: 2591
18, {8}, 1: 49393
18, 8, {1}: 845983
18, {8}, 5: 149
18, 8, {5}: 149
18, {8}, 7: 151
18, 8, {7}: 151
18, {8}, 11: 889211
18, 8, {11}: 2801
18, {8}, 13: 157
18, 8, {13}: 157
18, {8}, 17: 2753
18, 8, {17}: 578415690713087
18, {9}, 1: 163
18, 9, {1}: 163
18, {9}, 5: 167
18, 9, {5}: 167
18, {9}, 7: 1000357
18, 9, {7}: 3049
18, {9}, 11: 173
18, 9, {11}: 173
18, {9}, 13: 55579
18, 9, {13}: 3163
18, {9}, 17: 179
18, 9, {17}: 179
18, {10}, 1: 181
18, 10, {1}: 181
18, {10}, 7: 3968612127339681427
18, 10, {7}: 3373
18, {10}, 11: 191
18, 10, {11}: 191
18, {10}, 13: 193
18, 10, {13}: 193
18, {10}, 17: 197
18, 10, {17}: 197
18, {11}, 1: 199
18, 11, {1}: 199
18, {11}, 5: 3767
18, 11, {5}: 3659
18, {11}, 7: 3769
18, 11, {7}: 3697
18, {11}, 13: 211
18, 11, {13}: 211
18, {11}, 17: 3779
18, 11, {17}: 132239526911
18, {12}, 1: 140018322601
18, 12, {1}: 3907
18, {12}, 5: 74093
18, 12, {5}: 71699
18, {12}, 7: 223
18, 12, {7}: 223
18, {12}, 11: 227
18, 12, {11}: 227
18, {12}, 13: 229
18, 12, {13}: 229
18, {12}, 17: 233
18, 12, {17}: 233
18, {13}, 1: 4447
18, 13, {1}: 4231
18, {13}, 5: 239
18, 13, {5}: 239
18, {13}, 7: 241
18, 13, {7}: 241
18, {13}, 11: 4457
18, 13, {11}: 4421
18, {13}, 17: 251
18, 13, {17}: 251
18, {14}, 1: 4789
18, 14, {1}: 26565103
18, {14}, 5: 257
18, 14, {5}: 257
18, {14}, 11: 263
18, 14, {11}: 263
18, {14}, 13: 4801
18, 14, {13}: 4783
18, {14}, 17: 269
18, 14, {17}: 269
18, {15}, 1: 271
18, 15, {1}: 271
18, {15}, 7: 277
18, 15, {7}: 277
18, {15}, 11: 281
18, 15, {11}: 281
18, {15}, 13: 283
18, 15, {13}: 283
18, {15}, 17: 5147
18, 15, {17}: 30233087
18, {16}, 1: 32011489
18, 16, {1}: 30344239
18, {16}, 5: 293
18, 16, {5}: 293
18, {16}, 7: 5479
18, 16, {7}: 95713
18, {16}, 11: 5483
18, 16, {11}: 5393
18, {16}, 13: 32011501
18, 16, {13}: 5431
18, {16}, 17: 98801
18, 16, {17}: 5507
18, {17}, 1: 307
18, 17, {1}: 307
18, {17}, 5: 311
18, 17, {5}: 311
18, {17}, 7: 313
18, 17, {7}: 313
18, {17}, 11: 317
18, 17, {11}: 317
18, {17}, 13: 5827
18, 17, {13}: 9770144707511081415118442597789015238720654947319882836100223544506052645981243442054558121499672250712069138857313219
[/CODE]

sweety439 2020-12-13 10:49

[QUOTE=sweety439;566066]Consider the "simplest" families x{y} and {x}y, where x,y are base b digits

Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1

Zeros means there are no such prime of this family with <= 3000 digits (include the case that this family is ruled out to contain only composites)[/QUOTE]

For the zeros in the table: (for bases <=16)

Base 5:

{1}3: covering set {2,3}
{1}4: covering set {2,3}
3{1}: covering set {2,3}
4{1}: covering set {2,3}

Base 9:

{1}1: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8
1{1}: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8
{1}5: covering set {2,5}
2{7}: covering set {2,5}
3{1}: full algebra factors: (25*9^n-1)/8 = (5*3^n-1)*(5*3^n+1)/8
{3}5: covering set {2,5}
{3}8: covering set {2,5}
3{8}: full algebra factors: 4*9^n-1 = (2*9^n-1)*(2*9^n+1)
5{1}: covering set {2,5}
5{7}: covering set {2,5}
6{1}: covering set {2,5}
{7}2: covering set {2,5}
{7}5: covering set {2,5}
{8}5: full algebra factors: 9^n-4 = (3^n-2)*(3^n+2)

Base 11:

{1}3: covering set {2,3}
{1}4: covering set {2,3}
{1}9: covering set {2,3}
{1}A: covering set {2,3}
2{5}: covering set {2,3}
3{1}: covering set {2,3}
3{5}: covering set {2,3}
3{7}: covering set {2,3}
4{1}: covering set {2,3}
4{7}: covering set {2,3}
{5}2: covering set {2,3}
{5}3: covering set {2,3}
5{7}: (unsolved family) -------------------------------------------------------
{5}8: covering set {2,3}
{5}9: covering set {2,3}
{7}3: covering set {2,3}
{7}4: covering set {2,3}
{7}9: covering set {2,3}
{7}A: covering set {2,3}
8{5}: covering set {2,3}
9{1}: covering set {2,3}
9{5}: covering set {2,3}
9{7}: covering set {2,3}
A{1}: covering set {2,3}
A{7}: covering set {2,3}

Base 13:

{1}5: covering set {2,5,17}
{1}7: covering set {2,7}
{1}8: covering set {2,7}
2{9}: covering set {2,7}
{3}7: covering set {2,7}
{3}A: covering set {2,7}
4{B}: covering set {2,7}
{5}7: covering set {2,7}
{5}C: covering set {2,7}
7{1}: covering set {2,7}
7{3}: covering set {2,7}
7{5}: covering set {2,7}
7{9}: covering set {2,7}
7{B}: covering set {2,7}
8{1}: covering set {2,7}
{9}2: covering set {2,7}
9{5}: (unsolved family) -------------------------------------------------------
{9}7: covering set {2,7}
A{3}: covering set {2,7}
{B}4: covering set {2,7}
{B}7: covering set {2,7}
C{5}: covering set {2,7}

Base 14:

{2}5: covering set {3,5}
3{D}: covering set {3,5}
{4}9: covering set {3,5}
4{D}: (smallest prime is 4D[SUB]19698[/SUB] = 5*14^19698-1)
5{B}: covering set {3,5}
6{1}: covering set {3,5}
6{B}: covering set {3,5}
{8}3: covering set {3,5}
{8}5: covering set {3,5}
8{D}: partial algebra factors: 9*14^n-1 = (3*14^(n/2)-1)*(3*14^(n/2)+1) for even n, divisible by 5 for odd n
A{1}: covering set {3,5}
A{D}: covering set {3,5}
B{1}: partial algebra factors: (144*14^n-1)/13 = (12*14^(n/2)-1)*(12*14^(n/2)+1)/13 for even n, divisible by 5 for odd n
{B}5: covering set {3,5}
{D}3: covering set {3,5}
{D}5: partial algebra factors: 14^n-9 = (14^(n/2)-3)*(14^(n/2)+3) for even n, divisible by 5 for odd n

Base 16:

1{5}: full algebra factors: (4*16^n-1)/3 = (2*4^n-1)*(2*4^n+1)/3
{4}1: full algebra factors: (4*16^n-49)/15 = (2*4^n-7)*(2*4^n+7)/15
{4}D: covering set {3,7,13}
7{3}: full algebra factors: (36*16^n-1)/5 = (6*4^n-1)*(6*4^n+1)/5
8{1}: full algebra factors: (121*16^n-1)/15 = (11*4^n-1)*(11*4^n+1)/15
8{5}: full algebra factors: (25*16^n-1)/3 = (5*4^n-1)*(5*4^n+1)/3
{8}F: covering set {3,7,13}
8{F}: full algebra factors: 9*16^n-1 = (3*4^n-1)*(3*4^n+1)
{C}B: full algebra factors: (4*16^n-9)/5 = (2*4^n-3)*(2*4^n+3)/5
{C}D: full algebra factors: (4*16^n+1)/5 = (2*4^n-2*2^n+1)*(2*4^n+2*2^n+1)
D{1}: full algebra factors: (196*16^n-1)/15 = (14*4^n-1)*(14*4^n+1)/15
D{B}: (unsolved family) -------------------------------------------------------
E{1}: covering set {3,7,13}
{F}7: full algebra factors: 16^n-9 = (4^n-3)*(4^n+3)

sweety439 2020-12-15 14:04

1 Attachment(s)
This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)

format of file:

b,x,{y}: smallest prime of the form x{y} in base b
b,{x},y: smallest prime of the form {x}y in base b

such primes are generalized near-repdigit primes base b

already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b)

Also,

* in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits)
* in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits)
* in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits)
* in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits)
* in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits)
* in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits)
* in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits)
* in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits)
* in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits)
* in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits)
* in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits)
* in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits)
* in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits)
* in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits)
* in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits)
* in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits)
* in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits)
* in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits)
* in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits)
* in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits)

sweety439 2020-12-15 16:06

In fact, all these primes are minimal primes (start with 2 digits) base b:

* Smallest generalized repunit prime base b (if exists)
* The primes for all k<b for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] base b
* The primes for all k<b for [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] base b
* Smallest generalized near-repdigit primes base b of the form x{y} or {x}y for all (x,y) digit pair (if exists)

Since ....

* Generalized repunit numbers base b are 111...111 in base b
* k<b for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] base b are [k]000...0001 in base b
* k<b for [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] base b are [k-1][b-1][b-1][b-1]...[b-1][b-1][b-1] in base b
* Generalized near-repdigit numbers base b of the form x{y} or {x}y are [x][y][y][y]...[y][y][y] or [x][x][x]...[x][x][x][y] in base b

sweety439 2020-12-15 16:12

I think that the "minimal primes (start with 2 digits) problem" for all bases b>90 will never be proven (when searched to 1M or 1G or even 1T base b digits, and even when we allow probable primes in place of proven primes.), like CRUS S/R280, S/R511, S/R855, and S/R910 problems and the "Sierpinski/Riesel twin prime conjecture" (the conjecture that 237 is the smallest k divisible by 3 such that k*2^n+-1 are not twin primes for all n>=1), for more information, see [URL="https://mersenneforum.org/showpost.php?p=564315&postcount=57"]this post[/URL]

sweety439 2020-12-17 08:40

1 Attachment(s)
[QUOTE=sweety439;566267]This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)

format of file:

b,x,{y}: smallest prime of the form x{y} in base b
b,{x},y: smallest prime of the form {x}y in base b

such primes are generalized near-repdigit primes base b

already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)[/QUOTE]

extended data to base 36

sweety439 2020-12-26 03:14

[QUOTE=sweety439;566014]* Case (5,1):

** [B]51[/B] is prime, and thus the only minimal prime in this family.

* Case (5,3):

** [B]53[/B] is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)

*** Since 225, 255, [B]5205[/B] are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)

**** However, all numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime, therefore, there is no minimal primes in this family.

**** For the 5{0,5}25 family, since [B]500025[/B] and [B]505525[/B] are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)

***** However, 500525 is not prime, therefore, there is no minimal primes in this family.

***** The smallest prime of the form 5{5}25 is [B]555555555555525[/B]

***** The smallest prime of the form 5{5}025 is [B]55555025[/B]

***** For the 5{5}0025 family, since 55555025 is prime, we only need to consider the numbers 50025, 550025, 5550025, 55550025 (since any digit combo 5555 between (5,0025) will produce smaller primes)

****** However, none of them are primes, therefore, there is no minimal primes in this family.

***** The smallest prime of the form 5{5}0525 is [B]5550525[/B]

***** The smallest prime of the form 5{5}00525 is [B]5500525[/B]

***** For the 5{5}05025 family, since 55555025 is prime, we only need to consider the numbers 505025, 5505025, 55505025 (since any digit combo 555 between (5,05025) will produce smaller primes)

****** However, none of them are primes, therefore, there is no minimal primes in this family.

* Case (5,7):

** [B]57[/B] is prime, and thus the only minimal prime in this family.[/QUOTE]

* Case (6,3):

** Since 65, 13, 23, 53, 73, [B]643[/B] are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)

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

* Case (6,5):

** [B]65[/B] is prime, and thus the only minimal prime in this family.

* Case (6,7):

** Since 65, 27, 37, 57, [B]667[/B] are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Since 107, 117, 147, 177, 407, 417, 717, 747, [B]6007[/B], [B]6477[/B], [B]6707[/B], [B]6777[/B] are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes)

**** All numbers of the form 6{0}17 or 6{0}77 are divisible by 3, thus cannot be prime.

**** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes)

***** None of 6017, 6047, 6077 are primes.

**** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.

**** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes)

***** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.

***** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.


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