Remain k for Riesel CK=1:


Remain k for Riesel CK=2:


Remain k for Riesel CK=3:


Remain k for Riesel CK=4:

14: (proven)
29: (proven)
44: (proven)
59: (proven)
74: (proven)
89: (proven)
104: (proven)
119: (proven)
134: (proven)
149: (proven)
164: (proven)
179: (proven)
194: (proven)
209: (proven)
224: (proven)
239: (proven)
254: (proven)
269: 1,
284: (proven)
299: (proven)
314: (proven)
329: (proven)
344: (proven)
359: (proven)
374: (proven)
389: (proven)
404: (proven)
419: (proven)
434: (proven)
449: (proven)
464: (proven)
479: (proven)
494: (proven)
509: (proven)
524: (proven)
539: (proven)
554: (proven)
569: (proven)
584: (proven)
599: (proven)
614: (proven)
629: (proven)
644: (proven)
659: 3,
674: (proven)
689: (proven)
704: (proven)
719: (proven)
734: (proven)
749: (proven)
764: (proven)
779: (proven)
794: (proven)
809: (proven)
824: (proven)
839: (proven)
854: (proven)
869: (proven)
884: (proven)
899: (proven)
914: (proven)
929: (proven)
944: (proven)
959: (proven)
974: (proven)
989: (proven)
1004: (proven)
1019: 2,

Remain k for Riesel CK=5:

11: (proven)
23: (proven)
35: (proven)
47: (proven)
71: (proven)
83: (proven)
95: (proven)
107: (proven)
131: (proven)
143: (proven)
155: (proven)
167: (proven)
191: (proven)
203: (proven)
215: (proven)
227: (proven)
251: (proven)
263: (proven)
275: 4,
287: (proven)
311: (proven)
323: (proven)
335: (proven)
347: 3,
371: (proven)
383: (proven)
395: (proven)
407: (proven)
431: (proven)
443: (proven)
455: (proven)
467: (proven)
491: (proven)
503: (proven)
515: (proven)
527: (proven)
551: (proven)
563: (proven)
575: 3,
587: 3,
611: (proven)
623: (proven)
635: (proven)
647: 4,
671: (proven)
683: (proven)
695: (proven)
707: (proven)
731: 3,
743: (proven)
755: (proven)
767: (proven)
791: (proven)
803: (proven)
815: (proven)
827: (proven)
851: (proven)
863: (proven)
875: (proven)
887: (proven)
911: (proven)
923: (proven)
935: (proven)
947: (proven)
971: (proven)
983: (proven)
995: (proven)
1007: (proven)

Remain k for Riesel CK=6:

34: (proven)
69: (proven)
139: (proven)
174: (proven)
244: (proven)
279: (proven)
349: (proven)
384: 1,
454: (proven)
489: 5,
559: (proven)
594: (proven)
664: (proven)
699: 3,
769: (proven)
804: (proven)
874: (proven)
909: (proven)
979: (proven)
1014: (proven)

Remain k for Riesel CK=7:


Remain k for Riesel CK=8:

20: (proven)
41: (proven)
62: (proven)
125: (proven)
146: (proven)
188: (proven)
230: (proven)
272: (proven)
293: (proven)
307: (proven)
356: (proven)
377: (proven)
398: 7,
440: (proven)
461: (proven)
482: (proven)
538: (proven)
545: (proven)
566: (proven)
608: (proven)
650: 4,
692: (proven)
713: 1, 7,
762: (proven)
776: (proven)
797: (proven)
818: (proven)
860: (proven)
881: 1,
902: (proven)
965: (proven)
986: 6,
993: (proven)

Remain k for Riesel CK=9:

19: (proven)
39: (proven)
79: (proven)
99: (proven)
109: (proven)
159: (proven)
189: (proven)
199: (proven)
219: (proven)
229: (proven)
259: (proven)
309: (proven)
319: (proven)
339: (proven)
379: (proven)
399: (proven)
429: 5,
439: (proven)
459: (proven)
469: (proven)
499: 5,
519: (proven)
549: 6,
579: (proven)
589: (proven)
619: 6,
639: (proven)
669: (proven)
679: (proven)
709: (proven)
739: (proven)
759: 1,
789: (proven)
799: (proven)
819: (proven)
829: (proven)
859: (proven)
879: (proven)
919: (proven)
939: (proven)
949: (proven)
999: (proven)

Remain k for Riesel CK=10:

32: (proven)
65: (proven)
98: (proven)
197: (proven)
296: (proven)
362: (proven)
428: (proven)
560: (proven)
593: (proven)
626: (proven)
725: (proven)
758: (proven)
857: (proven)
890: (proven)
956: 1,
1022: (proven)

Remain k for Riesel CK=11:


Remain k for Riesel CK=12:

142: (proven)
285: (proven)
571: (proven)
714: (proven)
901: (proven)
1000: (proven)

Remain k for Riesel CK=13:

5: (proven)
27: (proven)
38: (proven)
53: (proven)
55: (proven)
77: (proven)
101: (proven)
111: (proven)
123: 11,
173: 11,
195: (proven)
221: 11,
223: 4,
245: (proven)
302: (proven)
317: 11,
341: (proven)
363: (proven)
365: (proven)
387: (proven)
391: (proven)
413: (proven)
437: 7,
447: (proven)
463: 7,
475: 4,
485: (proven)
531: (proven)
533: 11,
535: 4, 5,
548: 7,
557: (proven)
581: 2, 5,
605: (proven)
615: 12,
643: (proven)
653: 4,
677: (proven)
701: (proven)
727: 4, 8,
773: (proven)
783: 3,
811: (proven)
812: 4,
821: 7,
845: (proven)
867: 8,
893: 7, 9,
895: 7,
917: (proven)
941: (proven)
951: 1,
1013: (proven)

The reason for (k=27 is excluded from S8) (k=4 is excluded from S16) (k=1 is excluded from R4) (k=1 is excluded from R8) is the same as (k=1 is excluded from CRUS R14) (k=1 is excluded from CRUS R18) (k=1 is excluded from CRUS R20) (k=1 is excluded from CRUS R24)

Conjecture: For any integer triple (k,b,c) (k>=1, b>=2, c != 0, gcd(k,c) = 1, gcd(b,c) = 1) such that (k*b^n+1)/gcd(k+1,b-1) cannot be ruled out (by full numerical covering set, or by full algebraic covering set, or by partial numerical/partial algebraic covering set) as either only containing composites or containing only finitely many primes, then there are infinitely many integers n>=1 such that (k*b^n+1)/gcd(k+1,b-1) is prime.

[URL="https://docs.google.com/document/d/e/2PACX-1vTptXk4o1cdLQYGl-_gEiFtsvUYnG2oxR3Kl28Rfqh19Mreqrym00f0tT97GZA5LHQc_i7wgHppPnpz/pub"]https://docs.google.com/document/d/e/2PACX-1vTptXk4o1cdLQYGl-_gEiFtsvUYnG2oxR3Kl28Rfqh19Mreqrym00f0tT97GZA5LHQc_i7wgHppPnpz/pub[/URL]

Update the file of Riesel conjectures to include the newest test limit of R2

[URL="https://docs.google.com/document/d/e/2PACX-1vQqS_5lg1VlomDNNh0Pn2xrA8ECAus830jQK7zOjfAvMeUsJyTDhhiK-VWr_aVqjxRpAUXZqud2f_6r/pub"]https://docs.google.com/document/d/e/2PACX-1vQqS_5lg1VlomDNNh0Pn2xrA8ECAus830jQK7zOjfAvMeUsJyTDhhiK-VWr_aVqjxRpAUXZqud2f_6r/pub[/URL]

Update the file of Sierpinski conjectures to include the newest test limit of S30

[QUOTE=sweety439;565051]Conjecture: For any integer triple (k,b,c) (k>=1, b>=2, c != 0, gcd(k,c) = 1, gcd(b,c) = 1) such that (k*b^n+1)/gcd(k+1,b-1) cannot be ruled out (by full numerical covering set, or by full algebraic covering set, or by partial numerical/partial algebraic covering set) as either only containing composites or containing only finitely many primes, then there are infinitely many integers n>=1 such that (k*b^n+1)/gcd(k+1,b-1) is prime.[/QUOTE]
What's c? Why it matters? (you don't use it at all in the series).
Technically, you say the following: "any series which does not contain only finitely many primes will contain infinitely many primes"
Thank you Mr. Obvious. Where is the conjecture?

[QUOTE=LaurV;565824]What's c? Why it matters? (you don't use it at all in the series).
Technically, you say the following: "any series which does not contain only finitely many primes will contain infinitely many primes"
Thank you Mr. Obvious. Where is the conjecture?[/QUOTE]

It should be (k*b^n+c)/gcd(k+c,b-1) rather than (k*b^n+1)/gcd(k+1,b-1), I copied/pasted and forget to change.

This conjecture is: For any integer triple (k,b,c) (k>=1, b>=2, c != 0, gcd(k,c) = 1, gcd(b,c) = 1) such that (k*b^n+c)/gcd(k+c,b-1) cannot be ruled out (by full numerical covering set, or by full algebraic covering set, or by partial numerical/partial algebraic covering set) as either only containing composites or containing only finitely many primes, there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c,b-1) is prime (for the examples of (k*b^n+c)/gcd(k+c,b-1) can be ruled out as either only containing composites or containing only finitely many primes, see post [URL="https://mersenneforum.org/showpost.php?p=562388&postcount=1087"]#1087[/URL], also see page 12 of [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL])

e.g. these families are conjectured to contain infinitely many primes with integer n>=1:

* (k*2^n-1)/gcd(k-1,2-1) for all 1<=k<509203
* (k*3^n-1)/gcd(k-1,3-1) for all 1<=k<12119
* (k*4^n-1)/gcd(k-1,4-1) for all 1<=k<361 which is not square (squares k ruled out as only containing composites (k=1 has prime [I]only for[/I] n=2) by full algebra factors)
* (k*5^n-1)/gcd(k-1,5-1) for all 1<=k<13
* (k*6^n-1)/gcd(k-1,6-1) for all 1<=k<84687
* (k*7^n-1)/gcd(k-1,7-1) for all 1<=k<457
* (k*8^n-1)/gcd(k-1,8-1) for all 1<=k<14 which is not cube (cubes k ruled out as only containing composites (k=1 has prime [I]only for[/I] n=3) by full algebra factors)
* (k*9^n-1)/gcd(k-1,9-1) for all 1<=k<41 which is not square (squares k ruled out as only containing composites by full algebra factors)
* (k*10^n-1)/gcd(k-1,10-1) for all 1<=k<334
* (k*11^n-1)/gcd(k-1,11-1) for all 1<=k<5
* (k*12^n-1)/gcd(k-1,12-1) for all 1<=k<376 except 25, 27, 64, 300, 324 (these k ruled out as only containing composites by partial algebra factors)
* (k*2^n+1)/gcd(k+1,2-1) for all 1<=k<78557
* (k*3^n+1)/gcd(k+1,3-1) for all 1<=k<11047
* (k*4^n+1)/gcd(k+1,4-1) for all 1<=k<419
* (k*5^n+1)/gcd(k+1,5-1) for all 1<=k<7
* (k*6^n+1)/gcd(k+1,6-1) for all 1<=k<174308
* (k*7^n+1)/gcd(k+1,7-1) for all 1<=k<209
* (k*8^n+1)/gcd(k+1,8-1) for all 1<=k<47 which is not cube (cubes k ruled out as only containing composites (k=27 has prime [I]only for[/I] n=1) by full algebra factors)
* (k*9^n+1)/gcd(k+1,9-1) for all 1<=k<31
* (k*10^n+1)/gcd(k+1,10-1) for all 1<=k<989
* (k*11^n+1)/gcd(k+1,11-1) for all 1<=k<5
* (k*12^n+1)/gcd(k+1,12-1) for all 1<=k<521

[QUOTE=sweety439;565823][URL="https://docs.google.com/document/d/e/2PACX-1vQqS_5lg1VlomDNNh0Pn2xrA8ECAus830jQK7zOjfAvMeUsJyTDhhiK-VWr_aVqjxRpAUXZqud2f_6r/pub"]https://docs.google.com/document/d/e/2PACX-1vQqS_5lg1VlomDNNh0Pn2xrA8ECAus830jQK7zOjfAvMeUsJyTDhhiK-VWr_aVqjxRpAUXZqud2f_6r/pub[/URL]

Update the file of Sierpinski conjectures to include the newest test limit of S30[/QUOTE]

[URL="https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub"]https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub[/URL]

Update the file of Sierpinski conjectures to include the newest test limit of S53 k=4

Status for b<=128, k<=128:

[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=993593680"]Sierpinski[/URL]

[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=993593467"]Riesel[/URL]

using "full numerical covering set", "full algebraic covering set", partial numerical / partial algebraic covering set" instead of "covering set", "full algebra factors", "partial algebra factors"

[QUOTE=sweety439;566385]Status for b<=128, k<=128:

[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=993593680"]Sierpinski[/URL]

[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=993593467"]Riesel[/URL]

using "full numerical covering set", "full algebraic covering set", partial numerical / partial algebraic covering set" instead of "covering set", "full algebra factors", "partial algebra factors"[/QUOTE]

If two or three of "full numerical covering set", "full algebraic covering set", "partial numerical / partial algebraic covering set" occur simultaneously, then the order is "full numerical covering set" --> "full algebraic covering set" --> "partial numerical / partial algebraic covering set"

e.g.

8*125^n+1 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

8*125^n-1 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

(361*4^n-1)/3 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

343*8^n-1 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

(100*16^n-1)/3 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

81*1024^n-1 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

(49*9^n-1)/8 is both "full numerical covering set" and "full algebraic covering set", then we use "full numerical covering set"

4*14^n-1 is both "full numerical covering set" and "partial numerical / partial algebraic covering set", then we use "full numerical covering set"

4*29^n-1 is both "full numerical covering set" and "partial numerical / partial algebraic covering set", then we use "full numerical covering set"

(9*19^n-1)/2 is both "full numerical covering set" and "partial numerical / partial algebraic covering set", then we use "full numerical covering set"

4*9^n-1 is both "full algebraic covering set" and "partial numerical / partial algebraic covering set", then we use "full algebraic covering set"

(1*9^n-1)/8 is both "full algebraic covering set" and "partial numerical / partial algebraic covering set", then we use "full algebraic covering set"

9*4^n-1 is both "full algebraic covering set" and "partial numerical / partial algebraic covering set", then we use "full algebraic covering set"

(3*9^n-1)/8 is both "full algebraic covering set" and "partial numerical / partial algebraic covering set", then we use "full algebraic covering set"

The formula is:

Sierpinski problem base b (b>=2): (k*b^n+1)/gcd(k+1, b-1)
Riesel problem base b (b>=2): (k*b^n-1)/gcd(k-1, b-1)

The lowest k (k>=1) found to have a NUMERIC covering set for Sierpinski/Riesel problems bases 2<=b<=2500 and b = 4096, 8192, 16384, 32768, 65536:

[URL="https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt"]Sierpinski problems[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Riesel%20CK%20for%20bases%20up%20to%202048.txt"]Riesel problems[/URL]

All k (k>=1) below the lowest k found to have a NUMERIC covering set must have a prime including multiples of the base (MOB) but excluding the k which can be ruled out as composite for all n >= 1 (by full covering set with all or partial algebraic factors) (for the examples, see post [URL="https://mersenneforum.org/showpost.php?p=550059&postcount=871"]#871[/URL])

All n must be >= 1