SNOB Sierpinski conjectures and proofs

Started: Apr. 15, 2026
Last update: May 8, 2026

Compiled by Gary Barnes. Odd k<1048576 is based on work completed and coordinated by the Seventeen New or Bust (SNOB) project.

SNOB Sierpinski conjecture reservations

Abbreviations:  b = base, MOB = multiple of the base

SNOB Sierpinski conjectures are the same as the classical Sierpinski conjectures except that primes must be b^n > k.

For Sierpinski base 2 k<=271129 only, the following will apply:
1. Only k's that have a small prime where b^n <= k and n > 1 will have their primes shown and are tested with the SNOB project.
2. k's that do not have a prime where b^n <= k and n > 1 will not have their primes shown and are worked on by other projects.
   The k's must still be accounted for and are counted and shown separately.

Requirements and inclusions:
1. Conjectures must have a finite covering set and cannot be a MOB.
2. Primes must be b^n > k.
3. k-values that are a MOB are included in the conjectures but some may be excluded. See exclusion 4 below.

k-values will be excluded if any of the following conditions are met:
1. All n-values are covered by one trivial factor.
2. All n-values are covered by algebraic factors or a combination of algebraic and trivial factor(s).
3. Make generalized Fermat numbers (GFn's), i.e. q^m*b^n+1 where m>=0 and q is a root of the base.
4. k is a MOB and both k*b^(n-1)+1 and k*b^n+1 are composite, where n = floor [ logb(k) ]. They will have the same prime as k / b.

Testing for odd k<1048576 not done through other projects is coordinated at PrimeGrid Seventeen or Bust: A new Sierpinski problem.

Base Conjectured Sierpinski k Covering set k's that make a full covering set with all or partial algebraic factors Trivial k's (factor) Remaining k's to find prime
(n testing limit)		 Top 10 k's with largest first primes: k (n) Comments / accounting of all k's
2 78557 3, 5, 7, 13, 19, 37, 73   none 10 k's remaining at n>=10M.

SNOB; 5 k's:
23971 (10M)
45323 (10M)
50777 (10M)
50873 (10M)
76877 (10M)

Original base 2 project
(k's will not be tested by SNOB); 5 k's:
21181 (45.07M)
22699 (45.2M)
24737 (45.13M)
55459 (45M)
67607 (44.89M)		 SNOB only:
68633 (2715609)
71657 (1146175)
54953 (622065)
57377 (447439)
44243 (440969)
22249 (408602)
28831 (204580)
68221 (200944)
39527 (143055)
23873 (136733)		 Original project k's are worked on by PrimeGrid's Seventeen or Bust project. See k's and test limits at Seventeen or Bust stats.

k = 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, and 65536 are GFn's with no known prime.

all-ks-SNOB-sierp-base2.txt
 
2
2nd conjecture		 271129 3, 5, 7, 13, 17, 241   none 22 k's remaining at n>=5.1M.

SNOB; 7 k's:
83599 (6M)
97667 (5.1M)
113266 (5.1M)
129769 (5.1M)
131416 (5.1M)
149693 (5.1M)
225803 (5.1M)

Original base 2 projects
(k's will not be tested by SNOB); 15 k's:
79309 (39.3M)
79817 (39.59M)
91549 (29.57M)
131179 (29.41M)
152267 (39.28M)
156511 (39.71M)
163187 (29.44M)
200749 (29.4M)
209611 (29.4M)
222113 (39.27M)
225931 (39.55M)
227723 (29.41M)
229673 (30.13M)
237019 (39.04M)
238411 (29.41M)		 SNOB only:
96407 (2954495)
228226 (1542784)
160817 (756599)
165541 (627460)
97159 (523526)
192023 (507229)
221989 (351586)
145459 (272314)
130562 (214879)
248131 (204924)		 Only 78557<k<271129 are considered.

Original project prime k's are being worked on by PrimeGrid's Prime Sierpinski Problem project. See k's and test limits at Prime Sierpinski stats.

Original project composite odd k's are being worked on by PrimeGrid's Extended Sierpinski Problem project. See k's and test limits at Extended Sierpinski stats.

k = 131072 and 262144 are GFn's with no known prime.

all-ks-SNOB-sierp-base2-2nd-conj.zip
2
3rd conjecture		 271577 3, 5, 7, 13, 17, 241   none none - proven 271463 (1805)
271169 (1725)
271403 (985)
271339 (502)
271465 (486)
271201 (476)
271193 (433)
271333 (412)
271309 (350)
271477 (240)		 Only 271129<k<271577 are considered.

all-ks-SNOB-sierp-base2-3rd-conj.txt
2
4th conjecture		 322523 3, 5, 7, 13, 37, 73, 109   none 272341 (2.5M)
274699 (2.4M)
279767 (2.4M)
285601 (2.5M)
286037 (2.5M)
287393 (2.5M)
289171 (2.4M)
294181 (2.4M)
305063 (2.4M)
310339 (2.4M)
311573 (2.4M)		 279361 (1613712)
312121 (1109856)
273679 (1052058)
305147 (1030527)
285473 (530921)
281543 (440853)
308423 (395337)
312863 (293881)
301607 (229647)
287899 (223886)		 Only 271577<k<322523 are considered.

all-ks-SNOB-sierp-base2-4th-conj.txt
2
5th conjecture		 327739 3, 5, 7, 13, 17, 97, 257   none none - proven 327679 (24046)
324169 (15802)
325133 (5389)
322577 (4007)
326119 (3446)
326569 (3402)
322783 (3248)
326329 (3054)
325253 (2673)
325771 (1860)		 Only 322523<k<327739 are considered.

all-ks-SNOB-sierp-base2-5th-conj.txt
2
6th conjecture		 482719 3, 5, 7, 13, 17, 241   none 26 k's remaining at n>=2.4M.

See k's and test limits at SNOB Sierpinski 6th Base 2 remain.		 473567 (2437371)
340759 (2339350)
365221 (1767932)
363917 (1655731)
357271 (1370332)
447061 (1206128)
392479 (958886)
481727 (883059)
441923 (774725)
428657 (720223)		 Only 327739<k<482719 are considered.

all-ks-SNOB-sierp-base2-6th-conj.zip
2
7th conjecture		 575041 3, 5, 7, 13, 17, 241   none 484763 (2.4M)
491147 (2.4M)
499337 (2.5M)
502613 (2.4M)
510698 (1M)
515357 (2.5M)
517913 (2.4M)
532703 (2.4M)
536839 (2.4M)
538943 (2.4M)
545401 (2.4M)
548033 (2.4M)
553159 (2.4M)
558482 (1M)
561769 (2.4M)
566569 (2.4M)
571471 (2.4M)		 499561 (1759204)
520471 (1756052)
504061 (1714720)
518671 (1157008)
545971 (1082956)
501107 (1058835)
559789 (1030634)
504769 (839566)
499729 (725234)
506749 (574746)		 Only 482719<k<575041 are considered.

k = 524288 is a GFn with no known prime.

all-ks-SNOB-sierp-base2-7th-conj.txt
2
8th conjecture		 603713 3, 5, 7, 13, 17, 241   none 580831 (2.4M)
583189 (2.4M)
588317 (2.4M)
589021 (2.4M)
590033 (2.5M)
599011 (2.4M)		 599003 (1828141)
599513 (1282453)
575539 (431950)
584971 (266656)
588083 (244477)
590329 (155334)
578689 (66070)
588349 (51706)
593417 (43043)
593851 (35428)		 Only 575041<k<603713 are considered.

all-ks-SNOB-sierp-base2-8th-conj.txt
2
9th conjecture		 903983 3, 5, 7, 13, 17, 241   none 59 k's remaining at n>=1M.

See k's and test limits at SNOB Sierpinski 9th Base 2 remain.		 878029 (2420202)
836687 (2390667)
633481 (2069040)
867151 (1952104)
616909 (1899194)
852019 (1763242)
794867 (1702787)
609737 (1689147)
844457 (1688323)
751999 (1589870)		 Only 603713<k<903983 are considered.

all-ks-SNOB-sierp-base2-9th-conj.zip
2
10th conjecture		 934909 3, 5, 7, 13, 19, 73, 109   none 904489 (2.4M)
925907 (2.4M)
926371 (2.4M)		 923177 (611483)
923359 (541446)
911123 (479981)
924683 (421877)
907043 (305293)
908282 (301791)
910733 (200233)
928997 (145355)
922463 (141321)
911791 (129892)		 Only 903983<k<934909 are considered.

all-ks-SNOB-sierp-base2-10th-conj.txt
2
11th conjecture		 965431 3, 5, 7, 13, 17, 241   none 935723 (2.4M)
941492 (1M)
945572 (1M)
946879 (2.4M)
957977 (2.5M)
961099 (2.5M)
964673 (2.4M)		 960301 (430616)
944011 (372216)
943373 (304161)
942227 (215687)
963227 (196403)
951593 (159929)
961313 (155421)
959929 (141906)
956749 (95966)
936773 (93777)		 Only 934909<k<965431 are considered.

all-ks-SNOB-sierp-base2-11th-conj.txt
2
12th conjecture		 1259779 3, 5, 7, 13, 19, 73, 109   none 64 k's remaining at n>=1M.

See k's and test limits at SNOB Sierpinski 12th Base 2 remain.		 983027 (2176623)
1001419 (1675042)
1039127 (1193367)
1034809 (1077230)
1056019 (973810)
1016693 (963829)
1013657 (922163)
984173 (872129)
1191181 (859056)
1245169 (793326)		 Only 965431<k<1259779 are considered.

k = 1048576 is a GFn with no known prime.

all-ks-SNOB-sierp-base2-12th-conj.zip