A Sierpinski/Riesel-like problem

[sweety439]

R70 at n=46072, no other (probable) prime found

[sweety439]

S36 at n=92902, no (probable) prime found

[sweety439]

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

[sweety439]

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)

[sweety439]

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.

[sweety439]

GitHub link for this thread: https://github.com/xayahrainie4793/E...el-conjectures

[sweety439]

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.

[sweety439]

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

result file attached.

[sweety439]

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

[sweety439]

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 can’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

[sweety439]

The bases which have GFN or half GFN remain are (only exist in Sierpinski side):

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