Minimal set of the strings for primes with at least two digits - Page 11

sweety439 · 2021-01-08, 20:21

Quote:

Originally Posted by sweety439

base 13:

Code:

184: 2393
190: 917093711
196: 2549
202: 75001187
262: 3407
268: 45293
274: 46307
280: 615161
352: 10053473
358: 292031598119
382: 4967
388: 852437
406: 5279
412: 152972717
430: 5591
436: 5669
460: 5981
466: 13309427
[no known prime for S13 k=484, 2B3{0}1 is unsolved family in base 13]
490: 82811
538: 1181987
544: 1195169
574: 97007
580: 7541
616: 8009
622: 8087
652: 531856430093
658: 90710887636643
874: 264712843161629123
880: 148721
886: 11519
892: 11597
952: 12377
958: 2104727
964: 162917
970: 12611
1198: 34216079
1204: 2645189
1210: 15731
1216: 15809
1276: 80067107693
1282: 36615203
1288: 2829737
1294: 16823
1366: 652758153339229918146262170195611899412691250234661073608281535607441327245782154274226163627511856910242680997040239850166408242104102157590110059393064877741749052590786638302327896695517321882076385248095689607714279245947320040491619053213124101066946141643999231501774298529691499984819614064572813909251081765622950846989229615735750095118621279499641063473790506905622553548920945551313745769942678628213020977681299198153663054881959698284804196609230966508441216040759006656791607
1372: 17837
1396: 18149
1402: 12127883118972635782067
1420: 18461
1426: 18539
1444: 18773
1450: 41413451
1474: 19163
1480: 448255157756534441
1498: 34900531513476539
1504: 19553
1552: 20177
1558: 263303
1588: 2321529421116208423755077
1594: 269387
1630: 21191
1636: 21269
1666: 47582627
1672: 21737
1888: 821828298004735653739505427655167983139612951149032024600411489000613133621406169905393832026983244445467247164419724522787738176502646863025015739869100207704101192365412789485747731430993843624803368161051869616583490868130229284719241207489969244826144259775702398961583808046895149627729346180500126934284621986296138708001440861667299169
1894: 24623
1900: 1549888369901
1906: 4187483
1966: 4319303
1972: 333269
1978: 56493659
1984: 25793

484*13^15198+1 (2B30₁₅₁₉₇1) is prime!!!

This prime is likely the third-largest "base 13 minimal prime (start with base+1)" (there is a larger probable prime 80₃₂₀₁₇111, and there is an unsolved family in base 13: 9{5})

pinhodecarlos · 2021-01-08, 20:37

Have you thought about writing a book about all of this exciting stuff?

VBCurtis · 2021-01-08, 21:11

14 posts on a single topic in 36 hours. Are you trying to get banned? It's working.

sweety439 · 2021-01-09, 10:34

Quote:

Originally Posted by pinhodecarlos

Have you thought about writing a book about all of this exciting stuff?

I have a pdf file for the proof (not complete, continue updating), I will complete the proof for bases 7, 9, 12

sweety439 · 2021-01-09, 10:38

It is conjectured that for all simple families x{y}z cannot be proved as only contain composites (for numbers > base) in one of these four ways:

** Periodic sequence p of prime divisors with p(n) | (xyyy...yyyz with n y's)
** Algebraic factors (e.g. difference-of-squares factorization, difference-of-cubes factorization, sum-of-cubes factorization, difference-of-5th-powers factorization, sum-of-5th-powers factorization, Aurifeuillian factorization of x^4+4*y^4, etc.) of x{y}z
** The combine of the above two ways (like the case of {B}9B in base 12)
** Reduced to (b^(r*n+s)+1)/gcd(b+1,2), and r*n+s can never be power of 2 (like the case of 8{0}1 in base 128)

Then x{y}z contain primes (for numbers > base).

sweety439 · 2021-01-09, 10:53

The simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b are of the form (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), this number has algebra factors if and only if:

either

* there is an integer r>1 such that both a*b^n and -c are perfect rth powers

or

* a*b^n*c is of the form 4*m^4 with integer m

If (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebra factors, then it must be composite, the only exception is when it is either GFN (generalized Fermat number) base b or GRU (generalized repunit number) base b, in these two cases this number may be prime, the only condition is the n is power of 2 if it is GFN, and the n is prime if it is GRU

GFNs and GRUs are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), where Phi is cyclotomic polynomial) or Zsigmondy numbers Zs(n,b,1) (see Zsigmondy's theorem)

GFNs and GRUs in bases 2<=b<=36:

Code:

base         GFN family         GRU family
2            1{0}1              {1}
3            {1}2               {1}
4            1{0}1              1{3}, {2}3
5            {2}3               {1}
6            1{0}1              {1}
7            {3}4               {1}
8            2{0}1, 4{0}1       1{7}, 3{7}
9            {4}5               1{4}, {6}7
10           1{0}1              {1}
11           {5}6               {1}
12           1{0}1              {1}
13           {6}7               {1}
14           1{0}1              {1}
15           {7}8               {1}
16           1{0}1              1{F}, 7{F}, {A}B, 2{A}B
17           {8}9               {1}
18           1{0}1              {1}
19           {9}A               {1}
20           1{0}1              {1}
21           {A}B               {1}
22           1{0}1              {1}
23           {B}C               {1}
24           1{0}1              {1}
25           {C}D               1{6}, {K}L
26           1{0}1              {1}
27           1{D}E, 4{D}E       1{D}, 4{D}
28           1{0}1              {1}
29           {E}F               {1}
30           1{0}1              {1}
31           {F}G               {1}
32           2{0}1, 4{0}1, 8{0}1, G{0}1              1{V}, 3{V}, 7{V}, F{V}
33           {G}H               {1}
34           1{0}1              {1}
35           {H}I               {1}
36           1{0}1              1{7}, {U}V

Note: we do not include the case where the "ground base" of the GFNs or GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=36 are:

Code:

base         GFN family         GRU family
4                               {1}
8            1{0}1              {1}
9                               {1}
16                              {1}, 1{5}, {C}D
25                              {1}
27           {D}E               {1}
32           1{0}1              {1}
36                              {1}

Note: the "ground base" of the GFNs or GRUs need not to be b (when b is perfect power), it may be root of b, it may also be negative integer which is root of b

sweety439 · 2021-01-09, 21:18

These bases 2<=b<=1024 have unsolved families which are GFNs:

{31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024}

Such families are:

* 4:{0}:1, 16:{0}:1 for b = 32
* 12:{62}:63 for b = 125
* 16:{0}:1 for b = 128
* 36:{0}:1 for b = 216
* 24:{171}:172 for b = 343
* 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512
* 10:{0}:1, 100:{0}:1 for b = 1000
* 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024
* 1:{0}:1 for other even bases b
* {((b-1)/2)}:((b+1)/2) for other odd bases b

These bases 2<=b<=1024 have unsolved families which are GRUs:

{185, 243, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}

Such families are:

* 40:{121} for b = 243
* {1} for other bases b

carpetpool · 2021-01-09, 21:44

Here's a suggestion:

Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space!

The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes.

Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?

sweety439 · 2021-01-10, 12:09

Quote:

Originally Posted by carpetpool

Here's a suggestion:

Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space!

The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes.

Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?

I have made a pdf file about this project, see the attached file in post #114

Well, the proofs for base 2, 3, 4 are really trivial, but they are part of the project, I want to store these proofs, and the pdf file was made recently

sweety439 · 2021-01-10, 19:27

Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=12:

Code:

11
15
17
1B
25
27
31
35
37
3B
45
4B
51
57
5B
61
67
6B
75
81
85
87
8B
91
95
A7
AB
B5
B7
221
241
2A1
2B1
2BB
401
421
447
471
497
565
655
665
701
70B
721
747
771
77B
797
7A1
7BB
907
90B
9BB
A41
B21
B2B
2001
200B
202B
222B
229B
292B
299B
4441
4707
4777
6A05
6AA5
729B
7441
7B41
929B
9777
992B
9947
997B
9997
A0A1
A201
A605
A6A5
AA65
B001
B0B1
BB01
BB41
600A5
7999B
9999B
AAAA1
B04A1
B0B9B
BAA01
BAAA1
BB09B
BBBB1
44AAA1
A00065
BBBAA1
AAA0001
B00099B
AA000001
BBBBBB99B
B0000000000000000000000000009B
400000000000000000000000000000000000000077

sweety439 · 2021-01-10, 19:28

There are totally 106 minimal primes (start with 2 digits) in base 12, there are 77 such primes in base 10

2021-01-09, 10:38	#115
sweety439 Nov 2016 B04₁₆ Posts	It is conjectured that for all simple families x{y}z cannot be proved as only contain composites (for numbers > base) in one of these four ways: Periodic sequence p of prime divisors with p(n) \| (xyyy...yyyz with n y's) Algebraic factors (e.g. difference-of-squares factorization, difference-of-cubes factorization, sum-of-cubes factorization, difference-of-5th-powers factorization, sum-of-5th-powers factorization, Aurifeuillian factorization of x^4+4y^4, etc.) of x{y}z * The combine of the above two ways (like the case of {B}9B in base 12) ** Reduced to (b^(rn+s)+1)/gcd(b+1,2), and rn+s can never be power of 2 (like the case of 8{0}1 in base 128) Then x{y}z contain primes (for numbers > base). Last fiddled with by sweety439 on 2021-04-05 at 16:12

2021-01-09, 10:53	#116
sweety439 Nov 2016 2²×3×5×47 Posts	The simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b are of the form (ab^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), this number has algebra factors if and only if: either there is an integer r>1 such that both ab^n and -c are perfect rth powers or ab^nc is of the form 4m^4 with integer m If (ab^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebra factors, then it must be composite, the only exception is when it is either GFN (generalized Fermat number) base b or GRU (generalized repunit number) base b, in these two cases this number may be prime, the only condition is the n is power of 2 if it is GFN, and the n is prime if it is GRU GFNs and GRUs are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), where Phi is cyclotomic polynomial) or Zsigmondy numbers Zs(n,b,1) (see Zsigmondy's theorem) GFNs and GRUs in bases 2<=b<=36: Code: base GFN family GRU family 2 1{0}1 {1} 3 {1}2 {1} 4 1{0}1 1{3}, {2}3 5 {2}3 {1} 6 1{0}1 {1} 7 {3}4 {1} 8 2{0}1, 4{0}1 1{7}, 3{7} 9 {4}5 1{4}, {6}7 10 1{0}1 {1} 11 {5}6 {1} 12 1{0}1 {1} 13 {6}7 {1} 14 1{0}1 {1} 15 {7}8 {1} 16 1{0}1 1{F}, 7{F}, {A}B, 2{A}B 17 {8}9 {1} 18 1{0}1 {1} 19 {9}A {1} 20 1{0}1 {1} 21 {A}B {1} 22 1{0}1 {1} 23 {B}C {1} 24 1{0}1 {1} 25 {C}D 1{6}, {K}L 26 1{0}1 {1} 27 1{D}E, 4{D}E 1{D}, 4{D} 28 1{0}1 {1} 29 {E}F {1} 30 1{0}1 {1} 31 {F}G {1} 32 2{0}1, 4{0}1, 8{0}1, G{0}1 1{V}, 3{V}, 7{V}, F{V} 33 {G}H {1} 34 1{0}1 {1} 35 {H}I {1} 36 1{0}1 1{7}, {U}V Note: we do not include the case where the "ground base" of the GFNs or GRUs is either perfect power or of the form -4m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families for bases 2<=b<=36 are: Code: base GFN family GRU family 4 {1} 8 1{0}1 {1} 9 {1} 16 {1}, 1{5}, {C}D 25 {1} 27 {D}E {1} 32 1{0}1 {1} 36 {1} Note: the "ground base" of the GFNs or GRUs need not to be b (when b is perfect power), it may be root of b, it may also be negative integer which is root of b Last fiddled with by sweety439 on 2021-03-26 at 13:23*

2021-01-09, 21:18	#117
sweety439 Nov 2016 2²·3·5·47 Posts	These bases 2<=b<=1024 have unsolved families which are GFNs: {31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024} Such families are: * 4:{0}:1, 16:{0}:1 for b = 32 * 12:{62}:63 for b = 125 * 16:{0}:1 for b = 128 * 36:{0}:1 for b = 216 * 24:{171}:172 for b = 343 * 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512 * 10:{0}:1, 100:{0}:1 for b = 1000 * 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024 * 1:{0}:1 for other even bases b * {((b-1)/2)}:((b+1)/2) for other odd bases b These bases 2<=b<=1024 have unsolved families which are GRUs: {185, 243, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015} Such families are: * 40:{121} for b = 243 * {1} for other bases b Last fiddled with by sweety439 on 2021-03-25 at 10:46

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
generalized minimal (probable) primes	sweety439	sweety439	127	2021-02-24 02:35
Minimal set of the strings for primes with at least two digits	sweety439	Miscellaneous Math	6	2019-11-25 07:37
Digit strings containing primes	davar55	Puzzles	13	2018-03-15 14:46
Primes from powers of 2 strings.	Flatlander	Puzzles	40	2011-02-10 09:42
Strings of Digits	davar55	Puzzles	5	2008-11-02 00:08

2021-01-08, 20:37	#112
pinhodecarlos "Carlos Pinho" Oct 2011 Milton Keynes, UK 5·977 Posts	Have you thought about writing a book about all of this exciting stuff?

2021-01-08, 21:11	#113
VBCurtis "Curtis" Feb 2005 Riverside, CA 2×5×11×43 Posts	14 posts on a single topic in 36 hours. Are you trying to get banned? It's working.

2021-01-09, 21:44	#118
carpetpool "Sam" Nov 2016 2×163 Posts	Here's a suggestion: Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space! The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes. Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?

2021-01-10, 19:28	#121
sweety439 Nov 2016 2²·3·5·47 Posts	There are totally 106 minimal primes (start with 2 digits) in base 12, there are 77 such primes in base 10