A Sierpinski/Riesel-like problem - Page 84

sweety439 · 2020-07-25, 20:01

Done to base 2500

sweety439 · 2020-07-26, 15:34

Some CK in the text files in post #772have been fixed:

S970: 139823 --> 134552 (k = 134552 has covering set {3, 7, 44851})
S1150: >1M --> 755222 (k = 755222 has covering set {3, 7, 63031})
S1876: >1M --> 473969 (k = 473969 has covering set {3, 37, 31723})

R1150: >1M --> 190243 (k = 190243 has covering set {3, 7, 63031})
R1414: 64 --> 284 (k = 64 is unlikely to have a covering set, this error is because for this k, all n <= 1000 has a prime factor <= 10000)
R1876: 793972 --> 475846 (k = 475846 has covering set {3, 37, 31723})

The text file in post #772 have these errors is because I only tested the primes <= 10000 and the exponents <= 1000, but all these covering sets of these k has a prime > 10000, now I tested the primes <= 100000 and the exponents <= 2100, the fixed files are in the post #914

sweety439 · 2020-07-26, 15:36

See https://github.com/xayahrainie4793/E...f9596b6ce34d02 (Sierpinski) and https://github.com/xayahrainie4793/E...18a8665bfad2d0 (Riesel) for the updating.

sweety439 · 2020-07-26, 15:55

Reserve: (for k's > CK)

S4: k = 1238 & 1286
R4: k = 1159 & 1189

sweety439 · 2020-07-26, 15:59

See https://github.com/xayahrainie4793/f...el-conjectures for the status of the first 4 CK for the Sierpinski/Riesel conjectures for the bases b<=64 with smaller CK

sweety439 · 2020-07-26, 16:12

Quote:

Originally Posted by sweety439

Reserve: (for k's > CK)

S4: k = 1238 & 1286
R4: k = 1159 & 1189

and found 2 (probable) primes for R4:

(1159*4^5628-1)/3
(1189*4^3404-1)/3

the first 4 conjectures of R4 are all proven!!!

the 2 k's for S4 are still remain.

sweety439 · 2020-07-26, 16:43

Quote:

Originally Posted by sweety439

We can find the 2nd, 3rd, 4th, ... n such that (k*b^n+-1)/gcd(k+-1,b-1) is prime. (i.e. find the 2nd, 3rd, 4th, ... prime of the form (k*b^n+-1)/gcd(k+-1,b-1) for fixed k and fixed Sierpinski/Riesel base b)

For the n's such that (k*b^n+-1)/gcd(k+-1,b-1) is prime: (excluding MOB, since if k is multiple of the base (b), then k and k / b are from the same family, if k = k' * b^r, then the exponent n for this k is (the first number > r for k') - r, and the correspond prime for this k is the correspond prime for n = (the first number > r for k') for k')

S2:

k = 1: 1, 2, 4, 8, 16, ... (sequence is not in OEIS)
k = 3: A002253
k = 5: A002254
k = 7: A032353
k = 9: A002256
k = 11: A002261
k = 13: A032356
k = 15: A002258
k = 17: A002259
k = 19: A032359
k = 21: A032360
k = 23: A032361
k = 25: A032362
k = 27: A032363
k = 29: A032364
k = 31: A032365
k = 33: A032366
k = 35: A032367
k = 37: A032368
k = 39: A002269
k = 41: A032370
k = 43: A032371
k = 45: A032372
k = 47: A032373
k = 49: A032374
k = 51: A032375
k = 53: A032376
k = 55: A032377
k = 57: A002274
k = 59: A032379
k = 61: A032380
k = 63: A032381

S3:

k = 1: A171381
k = 2: A003306
k = 4: A005537
k = 5: 2, 6, 12, 18, 26, 48, 198, 456, ... (sequence is not in OEIS)
k = 7: 1, 9, 33, 65, 337, ... (sequence is not in OEIS)
k = 8: A005538
k = 10: A005539
k = 11: 1, 3, 21, 39, 651, ... (sequence is not in OEIS)
k = 13: 2, 14, 32, 40, 112, ... (sequence is not in OEIS)
k = 14: A216890
k = 16: 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, ... (sequence is not in OEIS)

S4:

k = 1: 1, 2, 4, 8, ... (sequence is not in OEIS)
k = 2: A127936
k = 3: 1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, ... (sequence is not in OEIS)
k = 5: 1, 3, 6, 12, 15, 18, 36, 72, 81, 84, 117, 522, 1023, 1083, 1206, ... (sequence is not in OEIS)
k = 6: 2, 20, 94, 100, 104, 176, 1594, ... (sequence is not in OEIS)
k = 7: A002255

S5:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: A058934
k = 3: 2, 6, 8, 62, 120, 186, 414, 764, ... (sequence is not in OEIS)
k = 4: A204322
k = 6: A143279
k = 7: (covering set {2, 3})
k = 8: 1, 1037, ... (sequence is not in OEIS)

S6:

k = 1: 1, 2, 4, ... (sequence is not in OEIS)
k = 2: A120023
k = 3: A186112
k = 4: A248613
k = 5: A247260
k = 7: 1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555 ... (sequence is not in OEIS)
k = 8: 4, 10, 16, 32, 40, 70, 254, ... (sequence is not in OEIS)

S10:

k = 1: 1, 2, ... (sequence is not in OEIS)
k = 2: A096507
k = 3: A056807
k = 4: A056806
k = 5: A102940
k = 6: A056805
k = 7: A056804
k = 8: A096508
k = 9: A056797
k = 11: A102975
k = 12: 2, 38, 80, 9230, 25598, 39500, ... (sequence is not in OEIS)
k = 13: A289051
k = 14: A099017
k = 15: 1, 4, 7, 8, 18, 19, 73, 143, 192, 408, 533, 792, 3179, 7709, 9554, 52919, 56021, 61604, ... (sequence is not in OEIS)
k = 16: A273002

R2:

k = 1: A000043
k = 3: A002235
k = 5: A001770
k = 7: A001771
k = 9: A002236
k = 11: A001772
k = 13: A001773
k = 15: A002237
k = 17: A001774
k = 19: A001775
k = 21: A002238
k = 23: A050537
k = 25: A050538
k = 27: A050539
k = 29: A050540
k = 31: A050541
k = 33: A002240
k = 35: A050543
k = 37: A050544
k = 39: A050545
k = 41: A050546
k = 43: A050547
k = 45: A002242
k = 47: A050549
k = 49: A050550
k = 51: A050551
k = 53: A050552
k = 55: A050553
k = 57: A050554
k = 59: A050555
k = 61: A050556
k = 63: A050557

R3:

k = 1: A028491
k = 2: A003307
k = 4: A005540
k = 5: 1, 3, 5, 9, 15, 23, 45, 71, 99, 125, 183, 1143, ... (sequence is not in OEIS)
k = 7: 2, 4, 6, 8, 16, 18, 28, 36, 52, 106, 114, 204, 270, 292, 472, 728, 974, ... (sequence is not in OEIS)
k = 8: A005541
k = 10: A005542
k = 11: 22, 30, 46, 162, ... (sequence is not in OEIS)
k = 13: 1, 5, 25, 41, 293, 337, 569, 1085, ... (sequence is not in OEIS)
k = 14: 1, 11, 16, 80, 83, 88, 136, 187, 328, 397, 776, 992, 1195, ... (sequence is not in OEIS)
k = 16: 1, 3, 9, 13, 31, 43, 81, 121, 235, 1135, 1245, ... (sequence is not in OEIS)

R4:

k = 1: (proven composite by full algebra factors)
k = 2: A146768
k = 3: A272057
k = 5: 1, 2, 4, 5, 6, 7, 9, 16, 24, 27, 36, 74, 92, 124, 135, 137, 210, 670, 719, 761, 819, 877, 942, 1007, 1085, ... (sequence is not in OEIS)
k = 6: 1, 3, 5, 21, 27, 51, 71, 195, 413, ... (sequence is not in OEIS)
k = 7: 2, 3, 5, 12, 14, 41, 57, 66, 284, 296, 338, 786, 894, ... (sequence is not in OEIS)

R5:

k = 1: A004061
k = 2: A120375
k = 3: 1, 2, 4, 9, 16, 17, 54, 64, 112, 119, 132, 245, 557, 774, 814, 1020, 1110, ... (sequence is not in OEIS)
k = 4: A046865
k = 6: A257790
k = 7: 1, 5, 11, 13, 15, 41, 61, 77, 103, 123, 199, 243, 279, 1033, 1145, ... (sequence is not in OEIS)
k = 8: 2, 4, 8, 10, 28, 262, 356, 704, ... (sequence is not in OEIS)

R6:

k = 1: A004062
k = 2: A057472
k = 3: A186106
k = 4: 1, 3, 25, 31, 43, 97, 171, 213, 273, 449, 575, 701, 893, ... (sequence is not in OEIS)
k = 5: A079906
k = 7: 1, 2, 3, 13, 21, 28, 30, 32, 36, 48, 52, 76, 734, ... (sequence is not in OEIS)
k = 8: 1, 5, 35, 65, 79, 215, 397, 845, ... (sequence is not in OEIS)

R10:

k = 1: A004023
k = 2: A002957
k = 3: A056703
k = 4: A056698
k = 5: A056712
k = 6: A056716
k = 7: A056701
k = 8: A056721
k = 9: A056725
k = 11: A111391
k = 12: 5, 3191, 3785, 5513, 14717, ... (sequence is not in OEIS)
k = 13: A056707
k = 14: 1, 2, 3, 4, 5, 16, 21, 23, 62, 175, 195, 206, 261, 347, 448, 494, 689, 987, 1361, 8299, 13225, 21513, 23275, ... (sequence is not in OEIS)
k = 15: 1, 2, 15, 22, 27, 33, 38, 473, 519, 591, 699, 2273, 2476, 2985, 6281, 6947, 11990, 16828, 17096, 26236, 33459, 34963, ... (sequence is not in OEIS)
k = 16: A056714

Also these OEIS sequences:

A078680: S2, k = index
A040076: S2, k = index (allow n=0)
A050412: R2, k = index + 1
A040081: R2, k = index (allow n=0)
A291437: S3, k = index * 2 (allow n=0)
A177330: R4, k = prime(index) if prime(index) == 1 mod 3, n = value/2; k = 2*prime(index) if prime(index) == 2 mod 3, n = (value-1)/2
A250204: S6, k = index, k != 4 mod 5
A250205: R6, k = index, k != 1 mod 5
A217377: R6, k = index * 5 + 1 (allow n=0)
A069568: R10, k = index * 9 + 1
A083747: R10, k = index * 9 + 1 (allow n=0)
A090584: R10, k = index * 3 + 1, k != 1 mod 9 (allow n=0)
A090465: R10, k = index + 1, k != 1 mod 3 (allow n=0)
A257459: R10, k = prime(index) * 9 + 1
A232210: R10, k = prime(index) * 3 + 1, k != 1 mod 9
A257461: R10, k = prime(index) + 1, k != 1 mod 3

sweety439 · 2020-07-28, 12:54

Update all primes for R102 and some primes for S80

sweety439 · 2020-07-30, 14:54

Extended Sierpinski problem base b:

Finding and proving the smallest k>=1 such that (k*b^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

Extended Riesel problem base b:

Finding and proving the smallest k>=1 such that (k*b^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)

sweety439 · 2020-07-31, 05:12

The reason for why S8 k=27, S16 k=4, R4 k=1, R8 k=1, R16 k=1, R27 k=1, R36 k=1 are excluded from the conjectures is the same as the reason for why in CRUS conjectures, R12 k=1, R14 k=1, R18 k=1, R20 k=1, R24 k=1 are excluded from the conjectures

The reason is although they have a prime, but they can have only this prime, thus excluded from the conjectures, a k-value with no covering set is included from the conjectures if and only if this k-value may have infinitely many primes

sweety439 · 2020-08-03, 23:47

Quote:

Originally Posted by sweety439

There are many Riesel case for k=64 (since 64 = 4^3 = 8^2, thus n == 0 mod 2 or n == 0 mod 3 have algebra factors:

Bases 619, 1322, 2025, 2728, 3431, 4134, 4837, 5540, 6243, 6946, 7649, 8352, 9055, 9758, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 37

Bases 429, 1816, 3203, 4590, 5977, 7364, 8751, ...: n == 1 mod 3: factor of 19, n == 2 mod 3: factor of 73

Bases 391, 2462, 4533, 6604, 8675, ...: n == 1 mod 3: factor of 19, n == 5 mod 6: factor of 109

Bases 159, 862, 1565, 2268, 2971, 3674, 4377, 5080, 5783, 6486, 7189, 7892, 8595, 9298, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 19

Bases 1232, 3933, 6634, 9335, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 73

Bases 936, 4969, 9002, ...: n == 1 mod 6: factor of 37, n == 5 mod 6: factor of 109

Bases 957, 2344, 3731, 5118, 6505, 7892, 9279, ...: n == 1 mod 3: factor of 73, n == 2 mod 3: factor of 19

Bases 1322, 4023, 6724, 9425, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 37

Bases 4315, ...: n == 1 mod 3: factor of 73, n == 5 mod 6: factor of 109

Bases 482, 2553, 4624, 6695, 8766, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 19

Bases 3098, 7131, ...: n == 1 mod 6: factor of 109, n == 5 mod 6: factor of 37

Bases 4079, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 73

Also, (for Riesel k=64)

Bases 235, 748, 1261, 1774, 2287, 2800, 3313, 3826, 4339, 4852, 5365, 5878, 6391, 6904, 7417, 7930, 8443, 8956, 9469, 9982, ...: n == 1 mod 3: factor of 3, n == 2 mod 3: factor of 19

Bases 397, 1396, 2395, 3394, 4393, 5392, 6391, 7390, 8389, 9388, ...: n == 1 mod 3: factor of 3, n == 5 mod 6: factor of 37

Bases 721, 2692, 4663, 6634, 8605, ...: n == 1 mod 3: factor of 3, n == 2 mod 3: factor of 73

Bases 1045, 3988, 6931, 9874, ...: n == 1 mod 3: factor of 3, n == 5 mod 6: factor of 109

Bases 334, 847, 1360, 1873, 2386, 2899, 3412, 3925, 4438, 4951, 5464, 5977, 6490, 7003, 7516, 8029, 8542, 9055, 9568, ...: n == 1 mod 3: factor of 19, n == 2 mod 3: factor of 3

Bases 307, 1306, 2305, 3304, 4303, 5302, 6301, 7300, 8299, 9298, ...: n == 1 mod 6: factor of 37, n == 2 mod 3: factor of 3

Bases 1468, 3439, 5410, 7381, 9352, ...: n == 1 mod 3: factor of 73, n == 2 mod 3: factor of 3

Bases 2008, 4951, 7894, ...: n == 1 mod 6: factor of 109, n == 2 mod 3: factor of 3

2020-07-26, 15:34	#915
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	Some CK in the text files in post #772have been fixed: S970: 139823 --> 134552 (k = 134552 has covering set {3, 7, 44851}) S1150: >1M --> 755222 (k = 755222 has covering set {3, 7, 63031}) S1876: >1M --> 473969 (k = 473969 has covering set {3, 37, 31723}) R1150: >1M --> 190243 (k = 190243 has covering set {3, 7, 63031}) R1414: 64 --> 284 (k = 64 is unlikely to have a covering set, this error is because for this k, all n <= 1000 has a prime factor <= 10000) R1876: 793972 --> 475846 (k = 475846 has covering set {3, 37, 31723}) The text file in post #772 have these errors is because I only tested the primes <= 10000 and the exponents <= 1000, but all these covering sets of these k has a prime > 10000, now I tested the primes <= 100000 and the exponents <= 2100, the fixed files are in the post #914 Last fiddled with by sweety439 on 2020-07-26 at 15:41

2020-07-31, 05:12	#923
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	The reason for why S8 k=27, S16 k=4, R4 k=1, R8 k=1, R16 k=1, R27 k=1, R36 k=1 are excluded from the conjectures is the same as the reason for why in CRUS conjectures, R12 k=1, R14 k=1, R18 k=1, R20 k=1, R24 k=1 are excluded from the conjectures The reason is although they have a prime, but they can have only this prime, thus excluded from the conjectures, a k-value with no covering set is included from the conjectures if and only if this k-value may have infinitely many primes Last fiddled with by sweety439 on 2020-08-06 at 06:44

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2020-07-26, 15:36	#916
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 B7A₁₆ Posts	See https://github.com/xayahrainie4793/E...f9596b6ce34d02 (Sierpinski) and https://github.com/xayahrainie4793/E...18a8665bfad2d0 (Riesel) for the updating.

2020-07-26, 15:55	#917
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2938₁₀ Posts	Reserve: (for k's > CK) S4: k = 1238 & 1286 R4: k = 1159 & 1189

2020-07-26, 15:59	#918
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·13·113 Posts	See https://github.com/xayahrainie4793/f...el-conjectures for the status of the first 4 CK for the Sierpinski/Riesel conjectures for the bases b<=64 with smaller CK

2020-07-30, 14:54	#922
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 5572₈ Posts	Extended Sierpinski problem base b: Finding and proving the smallest k>=1 such that (kb^n+1)/gcd(k+1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures) Extended Riesel problem base b: Finding and proving the smallest k>=1 such that (kb^n-1)/gcd(k-1,b-1) is not prime for all integers n>=1. (k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures)