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 2020-10-31, 04:53 #1079 sweety439     Nov 2016 17·149 Posts R70 at n=46072, no other (probable) prime found
 2020-10-31, 04:57 #1080 sweety439     Nov 2016 17×149 Posts S36 at n=92902, no (probable) prime found
 2020-10-31, 05:00 #1081 sweety439     Nov 2016 47458 Posts Update files * Update newest status for R6 k=1597 * Update newest status for R70 * Update newest status for S36 k=1814 * Fixed typo for S112 (the "Remaining k to find prime (n testing limit)" column, "other kl at n=6.9K" should be "other k at n=6.9K") Riesel conjectures Sierpinski conjectures
2020-10-31, 05:11   #1082
sweety439

Nov 2016

17·149 Posts

Quote:
 Originally Posted by sweety439 If k is rational power of base (b), then .... (let k = b^(r/s) with gcd(r,s) = 1) * For the Riesel case, this is generalized repunit number to base b^(1/s) * For the Sierpinski case, if s is odd, then this is generalized (half) Fermat number to base b^(1/s) * For the Sierpinski case, if s is even, then this is generalized repunit number to negative base -b^(1/s)
Let the divisor be d (i.e. d = gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel))

* For the Riesel case, this is generalized repunit number to base d+1
* For the Sierpinski case, if d = 1, then this is generalized Fermat number to base b^(1/s)
* For the Sierpinski case, if d = 2, then this is generalized half Fermat number to base b^(1/s)
* For the Sierpinski case, if d >= 3, then this is generalized repunit number to negative base -(d-1)

2020-10-31, 05:14   #1083
sweety439

Nov 2016

47458 Posts

Quote:
 Originally Posted by sweety439 Let the divisor be d (i.e. d = gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel)) * For the Riesel case, this is generalized repunit number to base d+1 * For the Sierpinski case, if d = 1, then this is generalized Fermat number to base b^(1/s) * For the Sierpinski case, if d = 2, then this is generalized half Fermat number to base b^(1/s) * For the Sierpinski case, if d >= 3, then this is generalized repunit number to negative base -(d-1)
If this is generalized (half) Fermat number, but the equation 2^x == r (mod s) has no solution, then this k has no possible prime, thus excluded from the conjecture, like the k proven composite by all or partial algebra factors.

 2020-11-04, 00:00 #1084 sweety439     Nov 2016 17·149 Posts GitHub link for this thread: https://github.com/xayahrainie4793/E...el-conjectures
2020-11-05, 13:15   #1085
sweety439

Nov 2016

17×149 Posts

Quote:
 Originally Posted by sweety439 S36 at n=92902, no (probable) prime found
S36 k=1814 passed n=100K, no (probable) primes, base released.

result file attached.
Attached Files
 S36 status.txt (13.3 KB, 6 views)

Last fiddled with by sweety439 on 2020-11-05 at 13:16

2020-11-05, 13:20   #1086
sweety439

Nov 2016

17·149 Posts

R70 passed n=50K, no other (probable) prime found, base released.

result file attached.
Attached Files
 R70 status.txt (48.5 KB, 6 views)

2020-11-06, 08:23   #1087
sweety439

Nov 2016

17·149 Posts

Quote:
 Originally Posted by sweety439 Like Bunyakovsky conjecture, it is conjectured that for all integer triples (k, b, c) satisfying these conditions: 1. k>=1, b>=2, c != 0 2. gcd(k, c) = 1, gcd(b, c) = 1 3. there is no finite set {p_1, p_2, p_3, ..., p_u} (all p_i (1<=i<=u) are primes) and finite set {r_1, r_2, r_3, ..., r_s} (all r_i (1<=i<=s) are integers > 1) such that for every integer n>=1: either (k*b^n+c)/gcd(k+c, b-1) is divisible by at least one p_i (1<=i<=u) or k*b^n and -c are both r_i-th powers for at least one r_i (1<=i<=s) or one of k*b^n and c is a 4th power, another is of the form 4*t^4 with integer t 4. the triple (k, b, c) is not in this case: c = 1, b = q^m, k = q^r, where q is an integer not of the form t^s with odd s > 1, and m and r are integers having no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution Then there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c, b-1) is prime.
Example of some (k,b,c) triple (k>=1, b>=2, c != 0, gcd(k, c) = 1, gcd(b, c) = 1) not satisfying these conditions:

* (k,b,c) = (78557,2,1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 19, 37, 73

* (k,b,c) = (271129,2,1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 17, 241

* (k,b,c) = (11047,3,1), in which all numbers are divisible by at least one of 2, 5, 7, 13, 73

* (k,b,c) = (419,4,1), in which all numbers are divisible by at least one of 3, 5, 7, 13

* (k,b,c) = (659,4,1), in which all numbers are divisible by at least one of 3, 5, 13, 17, 241

* (k,b,c) = (794,4,1), in which all numbers are divisible by at least one of 3, 5, 7, 13

* (k,b,c) = (7,5,1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (11,5,1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (174308,6,1), in which all numbers are divisible by at least one of 7, 13, 31, 37, 97

* (k,b,c) = (47,8,1), in which all numbers are divisible by at least one of 3, 5, 13

* (k,b,c) = (989,10,1), in which all numbers are divisible by at least one of 3, 7, 11, 13

* (k,b,c) = (5,11,1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (7,11,1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (521,12,1), in which all numbers are divisible by at least one of 5, 13, 29

* (k,b,c) = (4,14,1), in which all numbers are divisible by either 3 or 5

* (k,b,c) = (509203,2,-1), in which all numbers are divisible by at least one of 3, 5, 7, 13, 17, 241

* (k,b,c) = (334,10,-1), in which all numbers are divisible by at least one of 3, 7, 13, 37

* (k,b,c) = (1585,10,-1), in which all numbers are divisible by at least one of 3, 7, 11, 13

* (k,b,c) = (376,12,-1), in which all numbers are divisible by at least one of 5, 13, 29

* (k,b,c) = (919,4,-1), in which all numbers are divisible by at least one of 3, 5, 7, 13

* (k,b,c) = (13,5,-1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (17,5,-1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (14,8,-1), in which all numbers are divisible by at least one of 3, 5, 13

* (k,b,c) = (116,8,-1), in which all numbers are divisible by at least one of 3, 5, 13

* (k,b,c) = (148,8,-1), in which all numbers are divisible by at least one of 3, 5, 13

* (k,b,c) = (5,11,-1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (7,11,-1), in which all numbers are divisible by either 2 or 3

* (k,b,c) = (1,4,-1), in which all numbers factored as difference of squares

* (k,b,c) = (9,4,-1), in which all numbers factored as difference of squares

* (k,b,c) = (1,9,-1), in which all numbers factored as difference of squares

* (k,b,c) = (4,9,-1), in which all numbers factored as difference of squares

* (k,b,c) = (16,9,-1), in which all numbers factored as difference of squares

* (k,b,c) = (1,16,-1), in which all numbers factored as difference of squares

* (k,b,c) = (4,16,-1), in which all numbers factored as difference of squares

* (k,b,c) = (9,16,-1), in which all numbers factored as difference of squares

* (k,b,c) = (25,16,-1), in which all numbers factored as difference of squares

* (k,b,c) = (36,16,-1), in which all numbers factored as difference of squares

* (k,b,c) = (1,4,-9), in which all numbers factored as difference of squares

* (k,b,c) = (1,4,-25), in which all numbers factored as difference of squares

* (k,b,c) = (1,9,-4), in which all numbers factored as difference of squares

* (k,b,c) = (1,9,-16), in which all numbers factored as difference of squares

* (k,b,c) = (1,4,-25), in which all numbers factored as difference of squares

* (k,b,c) = (1,8,1), in which all numbers factored as sum of cubes

* (k,b,c) = (27,8,1), in which all numbers factored as sum of cubes

* (k,b,c) = (125,8,1), in which all numbers factored as sum of cubes

* (k,b,c) = (343,8,1), in which all numbers factored as sum of cubes

* (k,b,c) = (729,8,1), in which all numbers factored as sum of cubes

* (k,b,c) = (1,8,27), in which all numbers factored as sum of cubes

* (k,b,c) = (1,27,1), in which all numbers factored as sum of cubes

* (k,b,c) = (8,27,1), in which all numbers factored as sum of cubes

* (k,b,c) = (1,27,8), in which all numbers factored as sum of cubes

* (k,b,c) = (1,8,-1), in which all numbers factored as difference of cubes

* (k,b,c) = (27,8,-1), in which all numbers factored as difference of cubes

* (k,b,c) = (1,8,-27), in which all numbers factored as difference of cubes

* (k,b,c) = (125,8,-1), in which all numbers factored as difference of cubes

* (k,b,c) = (1,27,-1), in which all numbers factored as difference of cubes

* (k,b,c) = (8,27,-1), in which all numbers factored as difference of cubes

* (k,b,c) = (1,27,-8), in which all numbers factored as difference of cubes

* (k,b,c) = (1,32,1), in which all numbers factored as sum of 5th powers

* (k,b,c) = (1,32,-1), in which all numbers factored as difference of 5th powers

* (k,b,c) = (4,16,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (324,16,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (2500,16,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (4,81,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (4,256,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (4,625,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (64,81,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (64,256,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (64,625,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (324,256,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (324,625,1), in which all numbers factored as x^4+4*y^4

* (k,b,c) = (25,12,-1), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (64,12,-1), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (4,19,-1), in which even n factored as difference of squares and odd n is divisible by 5

* (k,b,c) = (9,14,-1), in which even n factored as difference of squares and odd n is divisible by 5

* (k,b,c) = (4,24,-1), in which even n factored as difference of squares and odd n is divisible by 5

* (k,b,c) = (9,24,-1), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (4,39,-1), in which even n factored as difference of squares and odd n is divisible by 5

* (k,b,c) = (9,34,-1), in which even n factored as difference of squares and odd n is divisible by 5

* (k,b,c) = (81,17,-1), in which even n factored as difference of squares and odd n is divisible by 2

* (k,b,c) = (144,28,-1), in which even n factored as difference of squares and odd n is divisible by 29

* (k,b,c) = (289,28,-1), in which even n factored as difference of squares and odd n is divisible by 29

* (k,b,c) = (16,33,-1), in which even n factored as difference of squares and odd n is divisible by 17

* (k,b,c) = (225,33,-1), in which even n factored as difference of squares and odd n is divisible by 2

* (k,b,c) = (289,33,-1), in which even n factored as difference of squares and odd n is divisible by 2

* (k,b,c) = (6,24,-1), in which odd n factored as difference of squares and even n is divisible by 5

* (k,b,c) = (27,12,-1), in which odd n factored as difference of squares and even n is divisible by 13

* (k,b,c) = (6,54,-1), in which odd n factored as difference of squares and even n is divisible by 5

* (k,b,c) = (76,19,-1), in which odd n factored as difference of squares and even n is divisible by 5

* (k,b,c) = (126,14,-1), in which odd n factored as difference of squares and even n is divisible by 5

* (k,b,c) = (300,12,-1), in which odd n factored as difference of squares and even n is divisible by 13

* (k,b,c) = (16,12,-49), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (441,12,-1), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (1156,12,-1), in which even n factored as difference of squares and odd n is divisible by 13

* (k,b,c) = (25,17,-9), in which even n factored as difference of squares and odd n is divisible by 2

* (k,b,c) = (1369,30,-1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 19

* (k,b,c) = (400,88,-1), in which even n factored as difference of squares and odd n is divisible by at least one of 3, 7, 13

* (k,b,c) = (324,95,-1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 229

* (k,b,c) = (3600,270,-1), in which even n factored as difference of squares and odd n is divisible by at least one of 7, 13, 37

* (k,b,c) = (93025,498,-1), in which even n factored as difference of squares and odd n is divisible by at least one of 13, 67, 241

* (k,b,c) = (61009,540,-1), in which even n factored as difference of squares and odd n is divisible by either 17 or 1009

* (k,b,c) = (343,10,-1), in which n divisible by 3 factored as difference of cubes and other n divisible by either 3 or 37

* (k,b,c) = (3511808,63,1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 37 or 109

* (k,b,c) = (27000000,63,1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 37 or 109

* (k,b,c) = (64,957,-1), in which n divisible by 3 factored as sum of cubes and other n divisible by either 19 or 73

* (k,b,c) = (2500,13,1), in which n divisible by 4 factored as x^4+4*y^4 and other n divisible by either 7 or 17

* (k,b,c) = (2500,55,1), in which n divisible by 4 factored as x^4+4*y^4 and other n divisible by either 7 or 17

* (k,b,c) = (16,200,1), in which n == 2 mod 4 factored as x^4+4*y^4 and other n divisible by either 3 or 17

* (k,b,c) = (64,936,-1), in which even n factored as difference of squares and n divisible by 3 factored as difference of cubes and other n divisible by either 37 or 109

* (k,b,c) = (8,128,1), in which the form equals 2^(7*n+3)+1 but 7*n+3 cannot be power of 2

* (k,b,c) = (32,128,1), in which the form equals 2^(7*n+5)+1 but 7*n+5 cannot be power of 2

* (k,b,c) = (64,128,1), in which the form equals 2^(7*n+6)+1 but 7*n+6 cannot be power of 2

* (k,b,c) = (8,131072,1), in which the form equals 2^(17*n+3)+1 but 17*n+3 cannot be power of 2

* (k,b,c) = (32,131072,1), in which the form equals 2^(17*n+5)+1 but 17*n+5 cannot be power of 2

* (k,b,c) = (128,131072,1), in which the form equals 2^(17*n+7)+1 but 17*n+7 cannot be power of 2

* (k,b,c) = (27,2187,1), in which the form equals (3^(7*n+3)+1)/2 but 7*n+3 cannot be power of 2

* (k,b,c) = (243,2187,1), in which the form equals (3^(7*n+5)+1)/2 but 7*n+5 cannot be power of 2

Last fiddled with by sweety439 on 2020-11-18 at 01:05

2020-11-06, 10:10   #1088
sweety439

Nov 2016

17·149 Posts

Quote:
 Originally Posted by sweety439 Like Bunyakovsky conjecture, it is conjectured that for all integer triples (k, b, c) satisfying these conditions: 1. k>=1, b>=2, c != 0 2. gcd(k, c) = 1, gcd(b, c) = 1 3. there is no finite set {p_1, p_2, p_3, ..., p_u} (all p_i (1<=i<=u) are primes) and finite set {r_1, r_2, r_3, ..., r_s} (all r_i (1<=i<=s) are integers > 1) such that for every integer n>=1: either (k*b^n+c)/gcd(k+c, b-1) is divisible by at least one p_i (1<=i<=u) or k*b^n and -c are both r_i-th powers for at least one r_i (1<=i<=s) or one of k*b^n and c is a 4th power, another is of the form 4*t^4 with integer t 4. the triple (k, b, c) is not in this case: c = 1, b = q^m, k = q^r, where q is an integer not of the form t^s with odd s > 1, and m and r are integers having no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution Then there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c, b-1) is prime.
This conjecture is that for any given integer triple (k,b,c) such that k>=1, b>=2, c != 0, gcd(k, c) = 1, gcd(b, c) = 1 and there is no obvious reason why there canâ€™t be a prime (or can be only prime for very small n, e.g. (4,16,1), (27,8,1), (1,4,-1), (1,16,-1), (1,27,-1), (1,36,-1), (1,128,-1), etc.) of the form (k*b^n+c)/gcd(k+c, b-1), then there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c, b-1) is prime (see page 12 of the article https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf), the obvious reason may be "full numeric covering set", "full algebraic covering set", or "partial numeric, partial algebraic covering set", see the section 4 "Some useful lemmas" of this pdf article or the section 5 "Polynomial factorization and partial factorization" of the pdf article https://www.utm.edu/staff/caldwell/preprints/2to100.pdf or the CRUS page http://www.noprimeleftbehind.net/cru...onjectures.htm and http://www.noprimeleftbehind.net/cru...onjectures.htm

 2020-11-06, 15:00 #1089 sweety439     Nov 2016 47458 Posts The bases which have GFN or half GFN remain are (only exist in Sierpinski side): Code: base,k 2,65536 6,1296 10,100 12,12 15,225 18,18 22,22 31,1 32,4 36,1296 37,37 38,1 40,1600 42,42 50,1 52,52 55,1 58,58 60,60 62,1 63,1 66,4356 67,1 68,1 70,70 72,72 77,1 78,78 83,1 86,1 89,1 91,1 92,1 93,93 97,1 98,1 99,1 104,1 107,1 108,108 109,1 117,117 122,1 123,1 124,15376 126,15876 127,1 128,16 135,1 136,136 137,1 138,138 143,1 144,1 147,1 148,148 149,1 151,1 155,1 161,1 166,166 168,1 178,178 179,1 182,1 183,1 186,1 189,1 192,192 193,193 196,196 197,1 200,1 202,1 207,1 211,1 212,1 214,1 215,1 216,36 217,217 218,1 222,222 223,1 225,225 226,226 227,1 232,232 233,1 235,1 241,1 243,27 244,1 246,1 247,1 249,1 252,1 255,1 257,1 258,1 262,262 263,1 265,1 268,268 269,1 273,273 280,78400 281,1 282,282 283,1 285,1 286,1 287,1 291,1 293,1 294,1 298,1 302,1 303,1 304,1 307,1 308,1 310,310 311,1 316,316 319,1 322,1 324,1 327,1 336,336 338,1 343,49 344,1 346,346 347,1 351,1 354,1 355,1 356,1 357,357 358,358 359,1 361,361 362,1 366,366 367,1 368,1 369,1 372,372 377,1 380,1 381,381 383,1 385,385 387,1 388,388 389,1 390,1 393,393 394,1 397,397 398,1 401,1 402,1 404,1 407,1 408,408 410,1 411,1 413,1 416,1 417,1 418,418 420,176400 422,1 423,1 424,1 437,1 438,438 439,1 443,1 446,1 447,1 450,1 454,1 457,457 458,1 460,460 462,462 465,465 467,1 468,1 469,1 473,1 475,1 480,1 481,481 482,1 483,1 484,1 486,486 489,1 493,1 495,1 497,1 500,1 509,1 511,1 512,2&4&16 514,1 515,1 518,1 522,522 524,1 528,1 530,1 533,1 534,1 538,1 541,541 546,546 547,1 549,1 552,1 555,1 558,1 563,1 564,1 570,324900 572,1 574,1 578,1 580,1 586,586 590,1 591,1 593,1 597,1 601,1 602,1 603,1 604,1 606,606 608,1 611,1 612,612 615,1 618,618 619,1 620,1 621,621 622,1 626,1 627,1 629,1 630,630 632,1 633,633 635,1 637,1 638,1 645,1 647,1 648,1 650,1 651,1 652,652 653,1 655,1 658,658 659,1 660,660 662,1 663,1 666,1 667,1 668,1 670,1 671,1 672,672 675,1 678,1 679,1 683,1 684,1 687,1 691,1 692,1 694,1 698,1 706,1 707,1 708,708 709,1 712,1 717,717 720,1 722,1 724,1 731,1 734,1 735,1 737,1 741,1 743,1 744,1 746,1 749,1 752,1 753,1 754,1 755,1 756,756 759,1 762,1 765,765 766,1 767,1 770,1 771,1 773,1 775,1 777,777 783,1 785,1 787,1 792,1 793,793 794,1 796,796 797,1 801,801 802,1 806,1 807,1 809,1 812,1 813,1 814,1 817,817 818,1 820,820 822,822 823,1 825,1 836,1 838,838 840,1 842,1 844,1 848,1 849,1 851,1 852,852 853,1 854,1 858,858 865,865 867,1 868,1 870,1 872,1 873,1 878,1 880,880 882,882 886,886 887,1 888,1 889,1 893,1 896,1 897,897 899,1 902,1 903,1 904,1 907,1 908,1 910,828100 911,1 915,1 922,1 923,1 924,1 926,1 927,1 932,1 933,933 937,1 938,1 939,1 941,1 942,1 943,1 944,1 945,1 947,1 948,1 953,1 954,1 958,1 961,1 964,1 967,1 968,1 970,970 974,1 975,1 977,1 978,1 980,1 983,1 987,1 988,1 993,1 994,1 998,1 999,1 1000,10 1002,1 1003,1 1005,1005 1006,1 1008,1008 1009,1 1012,1012 1014,1 1016,1 1017,1017 1020,1020 1024,4&16 Last fiddled with by sweety439 on 2020-11-06 at 15:07

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