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In the original Sierpinski (base 2) problem, the conjectured smallest Sierpinski number is 39565, 39565×2n+1 is divisible by at least one of the primes 3, 5, 7, 11, 17, 31, 61 for all n≥1, and there are 5 odd k below 39565 remaining with no known prime of the form k×2n+1:

10311, 11177, 12395, 28117, 3315E

(also 1 even k such that k+1 is prime, but without known prime for n≥1: 31E14)

The largest 10 known primes:

k n Dozenal length of the prime
5XEE X533545 2XE3E90
E181 4439XX2 1271647
14005 30X1289 X33X37
14555 2757621 894053
17591 2431000 7X7031
3127 1839086 57EE16
297E 1141XE3 388133
2783E 545X87 15EX60
33EE1 47999X 136X16
31E3E 40X837 1177X6
21657 400458 114940
22865 29807E 94865

A Colbert number is a prime >1000000 digits (i.e. ≥101000000) and has contributed to the in-progress computational proof that 39565 is the smallest Sierpinski number, there are only two known such numbers: E181×24439XX2+1 (1271647 digits) and 5XEE×2X533545+1 (2XE3E90 digits), similarly, a Colbert number base b is a prime >1000000 digits (i.e. ≥101000000) and has contributed to the in-progress computational proof the smallest generalized Sierpinski number base b (numbers k such that k×bn+1 is composite for all n≥1 and gcd(k+1,b−1) = 1 and k is not rational power of b (or this formula would be corresponding to generalized Fermat numbers)).

In the original Riesel (base 2) problem, the conjectured smallest Riesel number is 206817, 206817×2n−1 is divisible by at least one of the primes 3, 5, 7, 11, 15, 181 for all n≥1, and there are 41 odd k below 206817 remaining with no known prime of the form k×2n−1:

13E1, 5405, 11845, 1652E, 1X321, 23007, 32X11, 3728E, 3XX95, 4637E, 4826E, 52157, 5X655, 627X7, 6X947, 70995, 79685, 9380E, 9E29E, 9E41E, X4817, XXX61, E3747, EX135, 100837, 1329E5, 134X9E, 13760E, 138197, 13975E, 142E1E, 1464X7, 14761E, 155X15, 156327, 157207, 15739E, 15924E, 15E271, 16664E, 167X15, 171917, 179161, 195391, 1X8111, 1XX70E, 1E0467, 1E4EE1, 1EX387

(also 2 even k such that k−1 is prime, but without known prime for n≥1: 14E252 and 1E08E2)

The largest 10 known primes:

k n Dozenal length of the prime
112555 2EX9280 X01E08
202X11 24X42XE 807436
174E4E 249E080 805XE2
1E5E1 2344139 77709X
128067 22847X5 754539
172407 21961X3 72405X
101E3E 1X11786 61E600
15045E 154X524 4X31X4
6X185 1534126 49X05E
5E5E7 1335X35 432242
180397 131X9E5 428X7E
31E0E 1270392 409X54

Sierpinski 2nd problem[]

The conjectured 2nd Sierpinski number is 110XX1, and there are 14 odd k between 39565 and 110XX1 remaining with no known prime of the form k×2n+1:

39X91, 3X235, 44E91, 63XE7, 7414E, 766X7, 7X52E, 98211, 99381, X1377, X8655, XX8E7, XE94E, E0XE5, E51E7, E5E77

(also 2 even k such that k+1 is prime, but without known prime for n≥1: 4455X and 4XX6X)

The largest 10 known primes:

k n Dozenal length of the prime
81597 65X4600 1987799
49877 4840XX6 138705E
94325 3X039E7 10X0959
4447E 309X21E X33013
79241 24694X4 7E74XX
1055X5 19XX29E 613X31
109927 1763420 554261
X2277 10E637X 374719
46745 10X78X5 371794
96E85 EX3583 33835E
X8821 E58134 324X1E
71487 X36922 2X573X

Dual Sierpinski problem[]

All odd k below 39565 have at least one known (probable) prime of the form 2n+k.

The largest (probable) prime is 23065974+1E397 (for k = 1E397), it has X238XE digits.

The largest 10 (probable) primes:

k n Dozenal length of the (probable) prime
1E397 3065974 X238XE
20165 188221E 5928XE
1297 1650374 517X11
14555 90579E 263106
37735 6126EE 1850X9
4E3E 495553 14039X
2XE97 3E5284 112946
E181 22721X 75049
19E7E 12X861 41X25
31145 127E27 410XE
2E32E 85E55 24534
2XE8E 71943 1EE17

For 14 values of k, only non-proven strong probable primes are known (the largest proven prime is EE07-digit 236899+8625, and the smallest non-proven strong probable prime is 14901-digit 250072+X277, the current limit of primality proof (other than the numbers N such that either N−1 or N+1 has ≥40% factored) is 10000 digits).

Dual Riesel problem[]

Odd k below 206817 without known (probable) primes of the form 2nk

There are two definitions: allow negative primes or not.

Table of "Sierpinski, Riesel, dual Sierpinski, dual Riesel" problem for odd k up to 1000[]

k Sierpinski (smallest n≥1 such that k×2n+1 is prime) Riesel (smallest n≥1 such that k×2n−1 is prime) dual Sierpinski (smallest n≥1 such that 2n+k is prime) dual Riesel (smallest n≥1 such that |2nk| is prime)
1 1 2 1 2
3 1 1 1 3
5 1 2 1 1
7 2 1 2 1
9 1 1 1 1
E 1 2 1 2
11 2 3 2 1
13 1 1 1 1
15 3 2 1 2
17 6 1 2 1
19 1 1 1 1
1E 1 4 3 2
21 2 3 2 1
23 2 1 1 2
25 1 4 1 4
27 8 1 4 1
29 1 2 2 1
2E 1 2 1 2
31 2 1 2 3
33 1 3 1 1
35 1 2 1 2
37 2 7 2 1
39 2 1 1 1
3E 407 4 5 2
41 2 1 2 1
43 1 1 1 2
45 1 2 3 4
47 4 1 2 1
49 2 1 1 2
4E 5 10 1 4
51 4 3 8 1
53 1 2 2 1
55 1 4 1 2
57 2 5 2 3
59 1 1 1 1
5E 3 2 1 2
61 2 7 4 1
63 1 1 2 1
65 3 2 1 2
67 2 1 2 3
69 1 3 1 1
6E 1 2 7 2
71 4 5 2 1
73 2 1 1 2
75 1 4 3 4
77 8 1 4 1
79 2 3 2 2
7E 1 2 1 4
81 2 1 2 3
83 1 1 1 1
85 3 X 1 2
87 14 3 2 1
89 1 2 1 1
8E 3 X 1 2
91 6 9 2 1
93 1 2 1 1
95 1 8 7 2
97 2 1 4 1
99 3 1 5 2
9E 1 10 3 4
X1 8 1 4 3
X3 6 2 2 4
X5 1 2 1 4
X7 2 21 2 3E
X9 3 1 1 1
XE 1 2 3 2
E1 4 3 2 1
E3 1 1 1 2
E5 3 2 1 6
E7 2 1 X 1
E9 1 1 3 1
EE 45 2 3 2
101 6 5 2 3
103 8 1 1 3
105 3 4 1 8
107 4 5 4 1
109 1 3 2 1
10E 1 2 1 2
111 8 1 4 3
113 6 1 2 1
115 3 2 1 2
117 2 3 2 5
119 1 2 1 1
11E 7 4 5 2
121 2 1 2 1
123 8 2 1 2
125 1 2 3 4
127 2 1 2 1
129 2 1 1 2
12E 1 8 1 4
131 4 3 4 1
133 1 4 3 1
135 3 2 3 2
137 6 1 2 3
139 1 3 1 3
13E 1 16X 1 6
141 2 3 2 1
143 4 1 1 1
145 13 2 1 2
147 2 1 6 1
149 3 1 5 1
14E 11 2 3 2
151 2 1 6 3
153 2 2 2 3
155 1 4 1 4
157 18 1 4 5
159 2 4 4 1
15E 1 2 3 2
161 56 1 4 7
163 1 7 2 3
165 1 2 1 6
167 8 3 2 5
169 3 1 1 1
16E E 2 1 2
171 6 1 2 1
173 1 1 1 1
175 1 4 3 2
177 2 3 2 1
179 4 2 1 2
17E 1 4 1 4
181 30 E 4 1
183 1 2 3 1
185 1 4 5 2
187 6 E1 2 3
189 1 8 1 3
18E 1 6 5 8
191 2 5E 2 1
193 2 1 1 2
195 1E3 4 9 4
197 32 5 2 1
199 1 1 1 2
19E 25 2 3 6
1X1 2 E 2 1
1X3 2 4 1 2
1X5 3 8 1 8
1X7 4 1 18 1
1X9 1 2 2 1
1XE 7 6 1 2
1E1 2 5 2 3
1E3 2 1 1 1
1E5 1 2 1 2
1E7 26 E 6 1
1E9 1 1 3 1
1EE 3 4 25 2
201 X 1 2 3
203 4 2 1 3
205 1 2 7 4
207 2 5 4 1
209 3 1 4 2
20E 1 4 3 4
211 4 1 4 3
213 1 3 2 5
215 3 8 1 6
217 2 1 2 7
219 1 1 1 1
21E 9 22 1 2
221 4 3 2 1
223 1 2 1 1
225 7 6 5 2
227 2 3 6 1
229 1 1 4 2
22E 1 2 3 4
231 2 5 6 3
233 3 1 2 4
235 1 8 1 4
237 4 1 4 9
239 5 3 2 1
23E 17 16 1 2
241 4 1 4 55E
243 3 1 3 1
245 1 X 3 2
247 2 1E 2 5
249 1 4 1 3
24E 3 16 1 4
251 X 3 2 1
253 10 1 1 1
255 19 4 23 2
257 6 1 2 1
259 2 2 1 2
25E 1 4 3 8
261 24 3 8 1
263 1 2 2 2
265 5 2 1 4
267 10 1 4 3
269 1 4 2 1
26E 1 2 1 2
271 2 7 4 9
273 1 2 2 1
275 E X 1 2
277 12 1 2 5
279 3 1 1 1
27E 3849 2 13 2
281 8 1 2 1
283 2 1 1 2
285 E 18 3 4
287 4 5 8 1
289 1 2 2 2
28E 5 2 1 4
291 4 5 2 3
293 2 1 1 1
295 3 X 3 2
297 2 E 4 1
299 1 1 2 2
29E 3 2 1 8
2X1 2 3 X 3
2X3 1 1 3 1
2X5 1 4 3 2
2X7 6 1 2 5
2X9 2 2 1 3
2XE 1 4 1 8
2E1 4 7 8 1
2E3 2 8 3 1
2E5 7 2 3 2
2E7 2 1 2 3
2E9 1 1 1 3
2EE 1 2 1 6
301 2 3 4 1
303 2 4 2 1
305 1E 2 1 2
307 6 1 2 3
309 1 1 1 1
30E 1 6 7 2
311 4 3 2 1
313 2 2 1 2
315 3 94 3 4
317 10 3 4 1
319 1 2 2 2
31E 1 18 1 4
321 8 9 2 3
323 1 3 1 1
325 5 X 1 2
327 6 7 2 1
329 2 1 1 1
32E 17 2 5 2
331 2 1 X 1
333 3 1 3 2
335 1 6 7 4
337 2 5 2 3
339 6 1 1 4
33E 3 10 3 4
341 54 3 18 1
343 1 2 2 2
345 1 4 1 6
347 2 5 2 3
349 5 1 1 1
34E 1 3X 3 2
351 2 3 4 1
353 1 2 2 2
355 7 2 1 6
357 2 1 2 3
359 4 2 1 1
35E 5 2 7 2
361 4 1 2 1
363 2 1 1 2
365 1 8 5 9
367 8 1 14 1
369 2 6 3 2
36E 1 8 3 4
371 2 1 2 3
373 1 5 1 4
375 5 2 1 6
377 4 7 6 1
379 1 1 4 1
37E 3 4 9 2
381 6 3 6 3
383 1 1 4 3
385 11 2 3 8
387 2 1 6 5
389 3 5 2 4
38E 7 4 1 4
391 8 3 4 5
393 1 13 2 1
395 1 2 1 2
397 4 1 4 7
399 5 1 3 1
39E 1 2 9 2
3X1 2 3 2 5
3X3 2 1 1 3
3X5 3 8 9 4
3X7 2 1 2 1
3X9 1 2 1 2
3XE 9 2 3 4
3E1 4 1 2 1
3E3 2 2 1 2
3E5 13 4 1 8
3E7 4 3 4 1
3E9 2 3 2 1
3EE 1 4 1 2
401 2 1 4 3
403 6 E 3 1
405 1 16 5 2
407 2 3 2 9
409 1 2 1 3
40E 16E 2 5 4
411 2 7 2 1
413 3 1 1 2
415 1 2 3 4
417 2 3 2 1
419 2 1 1 2
41E 3 8 1 14
421 14 1 4 1
423 4 2 2 1
425 E 4 1 2
427 10 1 8 3
429 2 1 2 1
42E 1 6 1 2
431 6 3 2 9
433 1 1 1 1
435 3 2 1 2
437 2 1 6 1
439 3 3 5 1
43E 5 4 3 2
441 8 1 4 3
443 6 5 2 3
445 1 14 1 4
447 100 15 4 5
449 9 2 3 1
44E 3 2 3 2
451 2 9 2 7
453 1 1 1 3
455 1 6 1 6
457 4 3E 2 1
459 1 1 1 1
45E 13 X 11 2
461 1X 1 2 1
463 1 1 1 2
465 1 X 3 8
467 2 5 2 1
469 4 5 1 2
46E 1 327318 1 4
471 8 1 4 1
473 1 3 6 1
475 7 2 3 2
477 6 59 4 3
479 2 3 2 3
47E 7 2 1 6
481 2 7 2 5
483 5 2 1 1
485 3 2 5 2
487 6 3 2 1
489 3 1 1 2
48E 1 2 3 6
491 2 3 4 1
493 2 1 2 2
495 13 8 1 4
497 4 1 8 3
499 8 4 3 1
49E 9 4 5 2
4X1 2 5 2 7
4X3 1 3 1 3
4X5 5 2 3 12
4X7 8 3 4 1
4X9 3 1 2 2
4XE 3 4 1 4
4E1 2 59 6 3
4E3 1 2 3 1
4E5 1 2 17 2
4E7 2 1 2 5
4E9 3 1 1 3
4EE 1 4 3 8
501 8 5 8 1
503 1 3 2 2
505 1 6 1 4
507 2 1 4 3
509 1 5 2 1
50E 3 12 1 2
511 10 17 6 5
513 1 2 2 1
515 3 50 1 2
517 2 8E 2 9
519 1 1 1 1
51E 1 2 3 2
521 6 1 4 1
523 4 1 2 2
525 1 10 1 4
527 10 3 20 3
529 4 2 2 1
52E 1 2 1 2
531 4 5 2 E
533 2 4 1 1
535 1 1X 3 2
537 8 47 6 1
539 1 4 2 2
53E 1E 2 1 4
541 12 3 2 3
543 1 2 1 1
545 45 8 677 2
547 4 1 6 1
549 2 1 5 2
54E 1 14 3 8
551 4 3 4 3
553 1 3 2 5
555 1 44 1 4
557 8 5 X 7
559 1 3 3 1
55E 1 2 5 2
561 4 3 2 5
563 2 3 1 3
565 2E 2 5 6
567 6 1 6 1
569 5 1 3 2
56E 1 10 3 4
571 2 1 2 3
573 2 1 1 6
575 1 10 1 450
577 14 1 4 1
579 1 2 3 1
57E 3 2 3 2
581 6 9 2 3
583 3 1 1 3
585 3 6 1 6
587 6 E 2 1
589 2 2 1 1
58E 17 2 1 2
591 16 1 26 1
593 1 2 3 1
595 1 2 13 2
597 6 1 2 3
599 6 2 1 3
59E 5 8 7 4
5X1 4 7 4 1
5X3 2 2 4 2
5X5 3 8 3 4
5X7 2 1 4 3
5X9 1 1 2 8
5XE 7 12 1 6
5E1 2 3 2 5
5E3 3 1 1 1
5E5 3 4 1 2
5E7 X 3 2 1
5E9 1 1 1 1
5EE 5 4 7 2
601 2 9 4 1
603 2 1 4 2
605 5 4 3 4
607 8 1 4 3
609 1 2 2 4
60E 3 2 1 4
611 4 1 2 E
613 1 8 1 1
615 717 6 1 2
617 2 7 2 1
619 2 2 1 1
61E 7 2 5 2
621 2 1 6 1
623 1 3 4 2
625 1 2 7 4
627 2 1 4 3
629 3 5 5 4
62E 3 8 3 4
631 10 1 90 7
633 2 4 2 4
635 1 4 1 10
637 6 5 2 203
639 2 4 1 1
63E 1 2 3 2
641 16 E 4 1
643 1 2 2 2
645 23 6 1 6
647 2 3 6 3
649 3 6 3 1
64E 1 2 7 2
651 2 5 2 7
653 2 4 1 3
655 3 10 3 8
657 4 1 4 1
659 1 X 2 2
65E 1 2 1 4
661 8 1 2 3
663 1 1 1 1
665 3 6 7 2
667 6 1E 2 1
669 3 1 1 2
66E 3 6 9 6
671 2 3 2 1
673 4 1 1 2
675 1 12 E 4
677 2 3 4 1
679 4 1 6 2
67E 3 18 3 10
681 4 3 4 3
683 2 2 2 4
685 1 4 1 8
687 4 1 2 7
689 2 8 1 1
68E 9 6 9 2
691 4 7 2 1
693 1 1 1 2
695 3 2 5 X
697 2 5 2 1
699 5 2 1 2
69E 9 2 3 4
6X1 4 3 6 1
6X3 7 1 2 2
6X5 1 4 1 8
6X7 14 3 10 3
6X9 1 4 2 1
6XE 9 4 1 2
6E1 2 1 4 E
6E3 1 1 5 1
6E5 1 2 3 2
6E7 2 13 4 5
6E9 1 2 2 3
6EE 1E 2 1 4
701 6 1 2 5
703 3 3 1 1
705 1 2 3 2
707 8 1 2 1
709 4 9 1 2
70E 1 8 1 X
711 94 3 8 1
713 2 2 3 1
715 5 2 3 2
717 4 1 2 3
719 3 3 1 3
71E 1 6 1 6
721 2 3 4 1
723 4 1 2 1
725 3 8 1 2
727 2 3 6 3
729 1 1 3 1
72E 1 6 3 2
731 8 1 2 5
733 3 5 1 3
735 1 10 1 4
737 8 11 8 1
739 12 2 3 1
73E 1 2 3 2
741 2 1 2 3
743 5 4 1 3
745 9 2 1 6
747 2 27 8 1
749 1 1 2 1
74E 3 14 1 2
751 X 1 16 3
753 1 1 4 1
755 5 4 7 2
757 8 3 4 7
759 17 1 4 3
75E 7 14 3 4
761 10 1 4 5
763 6 3 2 5
765 3 2 1 4
767 2 21 2 2E
769 1 5 1 1
76E 5 2 1 2
771 2 27 2 1
773 3 4 1 1
775 7 4 5 2
777 2 19 2 1
779 1 3 1 2
77E 1 2672 7 4
781 2 3 2 1
783 4 1 1 2
785 35 18 3 4
787 10 1 8 1
789 5 2 2 2
78E 9 4 1 6
791 8 9 6 3
793 1 1 2 1
795 1 2 1 2
797 2 7 6 5
799 1 3 2 1
79E 13 2 1 2
7X1 2 7 6 5
7X3 3 2 5 1
7X5 1 4 E 2
7X7 30 1 4 5
7X9 2 1 4 3
7XE 9 4 5 4
7E1 4 1 10 5
7E3 1 4 3 8
7E5 3 12 3 4
7E7 4 1 2 7
7E9 2 1 1 5
7EE 3 2 1 6
801 16 7 6 1
803 1 1 3 1
805 3 6 9 2
807 2 5 2 3
809 7 2 1 3
80E 5 2 3 8
811 6 3 4 1
813 3 1 2 2
815 1 14 1 4
817 8 1 4 3
819 1 2 3 1
81E 1 8 7 2
821 6 5 2 7
823 3 1 1 3
825 7 2 5 6
827 2 3 2 1
829 1 3 1 2
82E 3 6 9 4
831 X 1 2 1
833 1 1 1 2
835 5 4 3 6
837 4 1 6 1
839 2 1 2 2
83E 1 4 1 10
841 680 5 4 3
843 2 3 8 1
845 1 1X 3 2
847 6 9 4 4E
849 6 1 2 3
84E 1 12736 1 12
851 28 17 2 5
853 2 3 1 1
855 7 4 5 2
857 2 1 2 1
859 3 1 1 2
85E 1 8 3 8
861 6 5 2 1
863 2 10 1 2
865 1 8 1 4
867 4 11 8 1
869 1 2 2 1
86E 5 4 1 2
871 4 1 6 3
873 2 1 6 1
875 3 6 3 2
877 2 23 4 11
879 5 4 2 3
87E 7 2 1 4
881 12 7 X 5
883 1 2 3 1
885 15 2 7 2
887 2 3 2 5
889 4 6 1 3
88E 5 4 5 10
891 4 1 4 1
893 6 2 4 2
895 1 2 5 4
897 10 5 4 3
899 1 3 3 5
89E 1 6 3 16
8X1 4 13 2 9
8X3 1 1 1 4
8X5 EE 2 1 6
8X7 1X 1 2 1
8X9 5 3 1 1
8XE 5 10 3 2
8E1 6 E 2 1
8E3 4 2 1 2
8E5 1 14 1 8
8E7 10 7 4 1
8E9 X 2 2 1
8EE 1 2 1 2
901 2 1 2 3
903 2 3 1 1
905 E 12 1 2
907 4 7 2 1
909 5 1 1 1
90E 3 2 39 2
911 2 1 6 1
913 5 1 3 2
915 5 22 3 4
917 2 15 2 3
919 8 1 1 4
91E 5 10 1 4
921 10 3 10 1
923 1 7 2 1
925 3 60 1 2
927 2 5 8 3
929 1 1 5 1
92E 1 2 7 2
931 2 3 X 5
933 1 5 5 3
935 13 2 15 4
937 X 1 26 5
939 1 12 5 6
93E 1 4 7 4
941 2 1 4 7
943 6 1 X 6
945 1 38 5 8
947 4 13 4 5
949 1 4 3 5
94E 1 2 7 6
951 4 1 2 7
953 1 5 1 5
955 3 2 7 6
957 14 3 2 1
959 1 1 1 2
95E 3 4 5 6
961 2 19 2 1
963 4 1 1 2
965 39 24 3 8
967 2 1 6 1
969 6 1 2 2
96E 5 4 1 8
971 10 3 8 3
973 1 2 4 1
975 E 4 E 2
977 4 5 6 7
979 2 1 6 3
97E 33 2 3 6
981 2 17 4 5
983 1 1 2 9
985 3 10 1 4
987 X 1 X 5
989 1 1 3 1
98E 9 4 7 2
991 4 3 2 5
993 3 4 1 3
995 1 4 E 8
997 8 3 4 1
999 2 2 4 2
99E 3 2 3 4
9X1 2 1 4 3
9X3 3 1 2 7
9X5 1 2 1 X
9X7 2 3 2 9
9X9 1 3 1 1
9XE 7 14 1 2
9E1 2 1 2 1
9E3 9 1 1 1
9E5 9 6 33 2
9E7 2 21 2 1
9E9 2 6 1 2
9EE 1 10 3 4
X01 4 3 8 1
X03 1 4 2 2
X05 13 2 1 4
X07 5X E1 2 3
X09 3 1 1 1
X0E 1 6 1 2
X11 2 2E 8 1
X13 2 1 2 1
X15 3 2 1 2
X17 6 1 6 3
X19 3 2 5 1
X1E 1 2 3 2
X21 2 3 4 5
X23 2 2 2 3
X25 1 8 1 4
X27 4 11 4 5
X29 5 3 3 1
X2E 3 4 3 2
X31 4 1 2 E
X33 3 1 1 3
X35 1 2 1 X
X37 8 3 2 1
X39 1 1 1 1
X3E 3 10 1 2
X41 16 7 2 1
X43 4 3 1 1
X45 5 4 7 2
X47 2 3 2 1
X49 X 2 1 2
X4E 1 14 5 4
X51 8 1 24 1
X53 4 2 3 2
X55 1 8 9 4
X57 2 9 2 3
X59 1 3 1 4
X5E 1 2 5 6
X61 2 7 14 1
X63 5 3 3 2
X65 13 2 5 6
X67 2X 1 2 3
X69 4 1 1 5
X6E 101 2 3 6
X71 2 1 X 1
X73 6 11 2 2
X75 9 14 1 14
X77 14 1 8 3
X79 1 2 4 1
X7E 3 X 3 2
X81 4 19 4 7
X83 1 4 2 3
X85 1 2 1 12
X87 2 3 4 5
X89 X 1 2 1
X8E 3 8 1 2
X91 2 3 2 E
X93 3 2 1 1
X95 5 2 27 2
X97 2 1 2 1
X99 2 5 1 2
X9E 1 4 3 4
XX1 4 1 10 1
XX3 10 3 2 2
XX5 11 16 1 4
XX7 2 49 2 3
XX9 2 1 1 1
XXE 3 X 3 2
XE1 4 3 6 1
XE3 2 2 2 2
XE5 2E 6 1 8
XE7 2 5 2 3
XE9 1 2 1 1
XEE 1 8 E 2
E01 12 1 4 1
E03 3 4 5 2
E05 3 4 3 50
E07 4 1 4 3
E09 1 3 2 X
E0E 1 2 1 4
E11 2 11 2 40E
E13 2 3 1 1
E15 1 12 3 2
E17 6 3 2 1
E19 2 1 1 2
E1E E 2 1 6
E21 X 1 2 1
E23 3 1 1 1
E25 9 2 3 2
E27 4 1 2 1
E29 2 8 1 2
E2E 3 4 1 10
E31 8 3 4 1
E33 4 2 2 1
E35 1 4 1 2
E37 8 1 X6 3
E39 1 1 3 1
E3E 3 6 5 2
E41 6 3 2 5
E43 1 6 1 3
E45 27 2 5 4
E47 X 15 X 1
E49 8 2 4 2
E4E X21 2 E 4
E51 2 3 6 3
E53 3 6 4 6
E55 1 40 3 10
E57 4 1 4 5
E59 1 7 2 4
E5E 3 2 1 8
E61 4 1 6 33
E63 1 5 2 1
E65 1 6 1 2
E67 2 E 2 9
E69 1 1 1 1
E6E 3 24 1 2
E71 172 5 6 1
E73 1 3 E 1
E75 1 2 7 2
E77 2 3 8 3
E79 2 9 4 3
E7E 1 4 E 4
E81 10 1 4 7
E83 2 3 4 4
E85 1 4 3 4
E87 4 1 X 7
E89 3 13 2 5
E8E 31 2 1 6
E91 14 7 2 9
E93 1 1 1 1
E95 3 8 1 2
E97 2E6 3 16 1
E99 13 2 3 1
E9E 1 6 7 2
EX1 4 5 2 3
EX3 2 1 1 3
EX5 E 14 5 4
EX7 8 3 10 1
EX9 4 4 3 2
EXE 3 14 3 4
EE1 2 1 2 3
EE3 5 3 1 10
EE5 5 2 1 6
EE7 6 7 6 1
EE9 5 1 3 1
EEE 267 2 5 2

Brier number[]

Numbers k which is both Sierpinski and Riesel.

CK=E0739280X466X9X4771997671

All (odd and even) k ≤ 1010 have at least one known prime of the form either k×2n+1 or k×2n−1, with n≥1

The Liskovets-Gallot Conjectures[]

[1]

[2]

Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n+1 or k×2n−1 is not prime for all even n or all odd n

In the "Riesel with even n" case, square k are excluded, since k×2n−1 has algebra factors for all even n

Sierpinski with even n[]

CK=32759, proven (only the k end with 3 or 9 are considered, the k end with 6 are "Sierpinski with odd n" for k/2, and the k end with 0 are "Sierpinski with even n" or "Sierpinski with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 primes:

k n Dozenal length of the prime
116X3 1303EE8 423774
2E269 103E4E6 3532XX
20879 378228 102279
17733 16291X 51037
17663 87160 24928
11933 51342 1511X
25743 462E4 13174
8599 44X60 12903
20179 3X782 1101X
2X033 39E64 109X7
10683 3620X E924
30843 35854 E777

Sierpinski with odd n[]

CK=47183, 3 k remain: 5443, 167E3, 268E9 (only the k end with 3 or 9 are considered, the k end with 6 are "Sierpinski with even n" for k/2, and the k end with 0 are "Sierpinski with even n" or "Sierpinski with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

k n Dozenal length of the prime
409X3 8E2095 25X8XX
41433 77190E 215122
2XE19 68X523 1X6846
3X693 1X6XE9 636X6
12359 152X09 49848
19E69 124E95 40308
34943 10E473 3741X
1X8X9 82107 23444
2E469 7X387 22387
44XE9 7963E 22112
25229 76759 21346
1853 52635 15539

Riesel with even n[]

CK=1E143, 2 k remain: 5613, 8389 (only the k end with 3 or 9 are considered, the k end with 6 are "Riesel with odd n" for k/2, and the k end with 0 are "Riesel with even n" or "Riesel with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

k n Dozenal length of the prime
E289 104X182 356298
E729 694504 1X8450
13413 80896 22E9E
15563 38192 10398
9073 32474 X862
13489 2291X 7575
1319 19E16 6151
6703 167E8 5261
5229 15XE2 4EE9
165E3 E4X0 3225
16233 997X 28X3
8723 65E8 1994

Riesel with odd n[]

CK=83E19, 4 k remain: 1XE73, 501X3, 75379, 7X753 (only the k end with 3 or 9 are considered, the k end with 6 are "Riesel with even n" for k/2, and the k end with 0 are "Riesel with even n" or "Riesel with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

k n Dozenal length of the prime
76259 9177E5 266EE8
71X03 7E1E73 226669
71573 3482E5 E4242
65649 331739 XXEE7
4013 2EX309 X0029
15443 25701E 83046
51689 1XE269 64932
70089 1944E1 5E63X
42159 1526E3 49763
4X13 131661 42786
83773 1176X1 39768
63603 1122X5 38187

Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n±1 are not both primes for all n≥1[]

CK=179, 9 k remain: 93, X3, 109, 113, 123, 133, 139, 163, 169 (only the k end with 3 or 9 are considered)

Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n+1 or k×2n−1 is not both primes for two consecutive n≥1[]

[3]

[4]

In the "Riesel" case, square k are excluded, since k×2n−1 has algebra factors for all even n

Sierpinski[]

CK=16959, 13X6 k remain, the smallest remain k are 43, 73, 79, 93, X3, X9, 159, 1E3, ... (only the k end with 3 or 9 are considered)

Riesel[]

CK=573, 28 k remain: 33, 133, 159, 163, 1X9, 1E3, 239, 253, 283, 289, 293, 2X9, 2E9, 333, 383, 393, 3X9, 3E3, 3E9, 413, 419, 423, 453, 489, 493, 499, 4X3, 503, 529, 533, 543, 563 (only the k end with 3 or 9 are considered)

Sierpinski problem base b[]

Links[]

[5]

[6]

[7]

k = 2 through 10 for bases up to 720

k = b−1 and b+1 for bases up to 720

Table for bases up to 20[]

All n must be ≥ 1.

k-values with at least one of the following conditions are excluded from the conjectures:

1. All n-values have a single trivial factor.

2. Make a full covering set with all or partial algebraic factors.

3. Make generalized Fermat numbers, i.e. qm×bn+1 where b is the base, m≥0, and q is a root of the base.

k-values that are a multiple of base (b) and where k+1 is composite are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

base conjectured smallest Sierpinski number covering set ks that make a full covering set with all or partial algebraic factors trivial ks number of remaining k remaining k top 6 primes comments
2 39565 {3, 5, 7, 11, 17, 31, 61} (none) 5 10311, 11177, 12395, 28117, 3315E 5XEE×2X533545+1
E181×24439XX2+1
14005×230X1289+1
14555×22757621+1
17591×22431000+1
3127×21839086+1
k = 31E14 is GFN with no known prime.
3 2029E39E01X {5, 7, 11, 15, 17, 31, 35, 141, 531} k = = 1 mod 2 (2) 314419 216X6X4, 2E98X24, 42X8792, 489X108, 4EE752X, 6167974, 72489XX, 8004E24, 8064188, 83210E4, E162E22, E21998X, EEE5XX4, 104E494X, 11257454, 13854068, 13E4X804, 1630552X, 17071EX4, 18356878, 18629454, 19278148, 1E33047X, 1E5X85XX, 1E642E88, 20718458, 215611E8, 21E4357X, 21E90XE8, 22435EX4, 231XX22X, 235X721X, 25602752, 2736E082, 28227482, 2887555X, 2943X62X, 2X251398, 31194428, 341689E8, 34237664, 352030XX, 358959X2, 35E24E0X, 36114618, 36662124, 38708X0X, 3903966X, 3X62XE28, 3XE55778, 3E754184, 40709444, 41721644, 41801E88, 4194X4X8, 42X64782, 434X2192, 4398E032, 44712458, 453E2EX4, 46E38214, 47540398, 49316222, 4X56667X, 4EX20X7X, 4EXX8572, 5056348X, 50578448, 518E4774, 5227X654, 52390828, 534071X0, 54093214, 54371708, 54715658, 547E5342, 54E29758, 5694X484, 56XX86XX, 57357838, 58X1X4X8, 59465X78, 5X466062, 5E182E94, 5EX45598, 61618288, 61942586, 62X76294, 6303495X, 640XE0XX, 64496568, 64E49664, 64E8107X, 6505X962, 65436434, 65E16X92, 67X535EX, 693E605X, 6944E64X, 699X950X, 6X23X708, 6X405E6X, 6X7XX8X2, 6X983E34, 70297378, 706X69E8, 70X1X202, 722X9X4X, 7422X958, 7452326X, 74860882, 75X35352, 76057662, 78934028, 7X392212, 7X581788, 8270814X, 843288E2, 850506E2, 85218542, 860484EX, 863753E8, 87000418, 89539E7X, 90XX074X, 9171E9X4, 9267X818, 9348X6X2, 93X59952, 94603676, 95190314, 95920288, 9598E694, 96791X1X, 97327X32, 9780801X, 9914X554, 9987XE52, 9X024X78, X1851268, X3035712, X315182X, X3928374, X39490X2, X40633E2, X408E7EX, X4892X34, X5X6X18X, X6575152, X6X08534, X6X2X094, X8371084, X9127902, XE7E12E4, XE835352, XEX41564, E0414172, E095X638, E1781672, E204X314, E2128524, E221E2X2, E33X4312, E39E0E16, E40E9664, E5542404, E5907072, E6291E38, E7598X74, E7618E7X, E8355102, E9263064, E9971438, EX28894X, EX5XX652, EE58E744, ..., 2028E621292, 2029183X27X, 20292148X02, 20292405058, 2029330905X, 20294138908, 202945512E2, 20295474368, 20295635874, 20296150E12, 20297828E44, 20299X8970X 14E97X938×32002EX+1
22X1576E2×31EX80E+1
28E4EE5X×31EX258+1
3X5X184X×31E83E2+1
22E32X274×31E65E5+1
E585E204×31E4972+1
4 32759 {5, 7, 11, 15, 181} k = = 2 mod 3 (3) 6 X886, 10311, 11177, 2476X, 28117, 313X6 E9EX×45277882+1
E181×4221XE51+1
2800X×41650744+1
28XXX×41389910+1
17591×41216600+1
3127×4X1X643+1
k = 31E14 is GFN with no known prime.
5 78702 {3, 7, 11, 27, 421} k = = 1 mod 2 (2) 27 3884, 4434, 639X, 13612, 1538X, 16428, 190X4, 201X2, 217E8, 21X82, 22584, 25774, 29E2X, 2XE4X, 3034X, 3122X, 33164, 33258, 35458, 37234, 38498, 406X8, 40938, 44148, 45922, 46124, 51048, 58748, 60EE2, 68678, 74378 681XX×5E18156+1
3E244×5X25X74+1
453EX×5875400+1
38728×5872469+1
752EX×584X3X0+1
6E444×5813EXX+1
6 84X58 {7, 11, 27, 31, 81} k = = 4 mod 5 (5) X 7793, 8489, 250E8, 382X1, 42988, 48263, 55327, 61X88, 80601, 816XX 5E9E9×6813E7X+1
68819×65188X0+1
1760X×637XX7X+1
6048X×6372E08+1
16178×6342148+1
2X52X×6317X91+1
k = 900, 4600, and 23000 are GFN with no known prime.
7 15E77XXX7X70 {5, 11, 17, 37, 61, 131, 141, 841} k = = 1 mod 2 (2)
k = = 2 mod 3 (3)
(not full tested) 3E7320, 659XE0, 781E9X, 82E8E6, 9E2004, 13229EX, 142080X, 1534E40, 1568920, 15XX440, 16E9956, 1728496, 17400E0, 223E154, 247193X, 2517020, 28X9794, 2X51574, 31119XX, 3138950, 3254380, 343E420, 3462316, 348EX74, 34E7820, 35420X0, 35471EX, 35X6886, 35E635X, 35E8680, 35EX126, 363124X, 36446X4, 3647050, 364904X, 3658400, 3677914, 3779864, 377X94X, 37X4626, 382463X, 384374X, 3886020, 3892880, 38E52E0, 3907760, 3949956, 3953566, 3970754, 39871X4, 39E1X94, 3X00476, 3X0X08X, 3X9022X, 3E0946X, 3E2931X, 4004304, 40053E6, 4013E74, 4030386, 40539XX, 406X5XX, 4084320, 4111000, 41X347X, 420X140, 4220366, 4220EEX, 42265E6, 4226944, 4292480, 4314926, 4323054, 4335934, 43904X4, 4404X1X, 44124EX, 441X3EX, 445980X, 44E3690, 4524536, 454080X, 455797X, 455X246, 4586E06, 459183X, 4596890, 45E3900, 461X72X, 462X9EX, 468556X, 4781X66, 4798500, 47X572X, 484711X, 4877544, 48993E6, 48E3984, 49260E6, 4933316, 4961450, 4962006, 4X34406, 4X34X14, 4X54200, 4X64214, 4X8XX2X, 4E23X70, 4E308XX, 4EE230X, 50233XX, 5029220, 504050X, 506524X, 50EX910, 5115446, 5176EX6, 521EE5X, 5222644, 5234E20, 5267016, 52X353X, 5333840, 535E446, 5375424, 5391584, 53E2260, 5406006, 5433E54, 5482014, 5491E46, 55016E4, 5516X9X, 553X636, 5620E4X, 566X200, 56795X6, 5690864, 56X6894, 5739104, 5761736, 5824704, 5840906, 58X9E3X, 5902X9X, 595X946, 5989504, 5990666, 5X50610, 5X77E30, 5XE01X4, 5E07386, 5E26824, 5E30146, 5E407X4, 5E5281X, 5E698E4, 5E8877X, 601524X, 604815X, 606963X, 6097296, 61018X6, 6134064, 617650X, 618418X, 6189390, 61E41E6, 624X366, 62790E4, 6303E74, 6346884, 6348380, 637687X, 6386540, 6391780, 6396E24, 63XEE86, 63E2500, 6404334, 6417256, 6478530, 6485364, 64X5756, 6521X74, 654X164, 6557626, 6622690, 6628606, 6674620, 6712990, 671E884, 674X114, 6753096, 67624E6, 6770140, 6774720, 6783184, 68052XX, 6876116, 6879654, 6885554, 68E128X, 69013XX, 6904470, 6917464, 6983574, 69E9150, 6X2001X, 6X26710, 6X98716, 6XE2814, 6E229E4, 6E3336X, 6E35970, 6E37266, 6E4X700, 6E80254, 70067X4, 7029860, 7033X96, 7040E36, 721EE4X, 7241E7X, 7288276, 72E5X74, 735E510, 737XX24, 738862X, 7395X56, 73X7696, 73E4446, 7406430, 74X821X, 74XX174, 752X474, 753E126, 754996X, 7572450, 7578594, 7592E54, 7598666, 7610880, 7630540, 768X176, 7717966, 773391X, 7734E86, 7751360, 7775326, 7792500, 781X844, 7831E20, 7868066, 7928410, 7968526, 7X14324, 7X36874, 7X65E4X, 7E13EXX, 7E5066X, 7E5498X, 7E5EE60, 7E89990, 8085006, 8099556, 80XE030, 811690X, 81X5604, 8240E46, 8265X7X, 8268E40, 826E080, 8294660, 8295006, 83031X0, 83036X6, 8358E16, 8376X90, 839355X, 842271X, 8436724, 8440550, 8475174, 8479X16, 8514364, 8521616, 8522084, 85X8E24, 863790X, 864155X, 865607X, 86753X6, 86X2536, 8700314, 8701XX0, 87EX300, 88218X4, 8848034, 89955EX, 8X034XX, 8X14074, 8X51190, 8X938X4, 8XEX136, 8E1EX2X, 8E8232X, 8E98956, 8E99574, 8EX03E4, 903258X, 903EX1X, 9043544, 9057346, 907E370, 9081946, 90824X0, 911E1E4, 911E410, 914E896, 9160E16, 91XX1X0, 9225394, 9241E6X, 9271E44, 9274XEX, 92E46X0, 92E506X, 9372430, 9376EE6, 9391970, 9413E90, 9443X54, 944E7X4, 9455454, 9459106, 9532306, 9541410, 954E600, 9552X84, 955621X, 9626434, 9649016, 964X134, 96E10E0, 972382X, 9733980, 975109X, 9757514, 9771960, 9773274, 9773666, 97791X0, 977E214, 9819EX4, 9844836, 995672X, 9971E56, 9988910, 9997364, 99X2E36, 99X7694, 99E4686, 9X5X220, 9X6977X, 9X775X4, 9X9327X, 9XX0816, 9XX5804, 9XEE98X, 9E25086, 9E2E4XX, 9E3513X, 9EE1254, 9EE8090, X012046, X0205EX, X045E46, X05023X, X0685E6, X0X5396, X101040, X155E0X, X159E20, X1874X0, X1XX254, X2856X0, X3X6E34, X3E7130, X3EE264, X421E5X, X432790, X4661E0, X4X1986, X5X3586, X5E8240, X613296, X678860, X69E93X, X724726, X77E4XX, X7X2670, X7XE080, X819126, X851684, X863534, X864830, X872420, X8785X0, X8E827X, X920986, X933480, X9X6834, XX00X94, XX16574, XX59E50, XE82406, XE95XE0, XE98820, XEX0034, XEX074X, XEX87EX, E06E070, E070E3X, E091066, E09X986, E157130, E166306, E172X86, E21665X, E23197X, E240076, E247780, E3326X6, E34464X, E34E7X6, E360160, E382300, E39642X, E3X7770, E468X34, E4E7E34, E501966, E5112EX, E5241X6, E53E676, E621E56, E69X360, E70X976, E763626, E767666, E7X285X, E7E5570, E7E756X, E806744, E834346, E85025X, E86X934, E929680, E935300, EX17296, EX91590, EX9341X, EE0X0E6, EE2X09X, EE36650, EE87726, EE9E20X, EEEX236, ..., 7X1X20×7121920+1
19XE344×7114432+1
1846756×71076X9+1
1546E04×7106X61+1
157147X×7E8311+1
6X6626×7E7292+1
8 3E {3, 5, 11} All k = m^3 for all n; factors to:
(m*2^n + 1) *
(m^2*4^n - m*2^n + 1)
k = = 6 mod 7 (7) 0 (proven) none (proven) 27×818+1
3X×84+1
34×84+1
31×84+1
24×84+1
14×84+1[1]
k = 1 and 8 proven composite by full algebraic factors.
9 1434 {5, 7, 11, 61} k = = 1 mod 2 (2) 1 1218 109X×931X00+1
1064×921633+1
1110×99391+1
90X×91E52+1
642×91071+1
1016×9920+1
X 5387 {7, E, 11, 31} k = = 2 mod 3 (3) 1 452X 2XE0×X40726+1
4350×X21E36+1
48XX×X10289+1
2431×X6EEE+1
4629×X69X9+1
36X4×X6270+1
k = 84 and 6E4 are GFN with no known prime.
E X42 {3, 7, 17, 31} k = = 1 mod 2 (2)
k = = 4 mod 5 (5)
0 (proven) none (proven) 67X×E125E14+1
X24×E13242+1
2X8×E7459+1
732×E1X29+1
9X4×E1598+1
442×E6X7+1
10 375 {5, 11, 25} k = = X mod E (E) 0 (proven) none (proven) 298×102X5626+1
276×101470+1
199×10458+1
29E×10267+1
256×10203+1
31×10147+1
k = 10 and 100 are GFN with no known prime.
11 E0 {5, 7, 15} k = = 1 mod 2 (2)
k = = 2 mod 3 (3)
0 (proven) none (proven) 40×113763+1
X0×11X94+1
8X×1148+1
54×1122+1
94×1110+1
9X×11E+1
12 4 {3, 5} k = = 10 mod 11 (11) 0 (proven) none (proven) 1×122+1[2]
3×121+1
2×121+1
13 X292X07695290 {11, 15, 95, 157, 181, X41, 22E1} k = = 1 mod 2 (2)
k = = 6 mod 7 (7)
(not full tested) X4808, 1854E6, 290X70, 794744, E46828, 1056556, 1328876, 1848708, 1858798, 1942008, 195E776, 1977000, 197EE38, 19X2962, 1X727X6, 1X729EX, 1E97990, 202537X, 2032680, 2052134, 2069870, 20E0958, 215X10X, 21X5966, 2211138, 2306784, 2311068, 2575690, 264E578, 27222E4, 276E740, 28080X4, 2832618, 2962668, 296685X, 2978400, 2997482, 29E3XE4, 2X18338, 2X64468, 2E05588, 3046484, 30X529X, 30E2152, 314X2E8, 3170574, 3251X72, 3252E6X, 330917X, 343X258, 3445446, 345E108, 34755E8, 34X5756, 3520560, 361627X, 36197XX, 3625X24, 3630X70, 3692512, 37E27E0, 385E642, 3875616, 3913X18, 397301X, 4000510, 4011EX2, 413X3XX, 4141886, 4152822, 42285E8, 4283202, 4325590, 4381916, 43E3X40, 4499366, 4503042, 4532864, 4541366, 461X45X, 47305X0, 4745306, 4916848, 49178E6, 4920144, 4930916, 4941540, 4952258, 4957280, 4X41654, 4X57108, 4X68610, 4X9XX36, 4XX23X2, 4XE92X0, 4EEE806, 5007E72, 50E72X0, 51208X4, 5149X28, 517010X, 5196374, 5210EE8, 5263212, 531E878, 5361222, 5427X34, 54383X0, 545XEEX, 548012X, 578X866, 5792818, 5877784, 5886970, 592060X, 5975764, 5976856, 5X12966, 6036086, 6077448, 6103568, 611X3EX, 6144924, 61768E2, 6243724, 6317412, 6326774, 6360572, 639EX38, 63X9110, 6434202, 6454570, 6455X3X, 64E7940, 64E8082, 6517556, 6568190, 664E490, 67021E8, 6889EE4, 68XE920, 68E0E16, 6967376, 69E349X, 6X3X338, 6X48968, 6X4E834, 6E77514, 6E87102, 6E9X882, 7011788, 702466X, 705278X, 70E2424, 7108184, 7279196, 72E6XEX, 7370156, 7380E66, 748E326, 7524326, 7620EE2, 771777X, 776X790, 777E416, 7790X02, 7816232, 78EX244, 7958640, 7X261E2, 7XE3924, 7E58E0X, 7EX5678, 8026X24, 803X1X0, 8127980, 8153840, 8195E98, 8369004, 8413930, 847E966, 85E8E1X, 862X924, 8685264, 8798116, 8850XX2, 8910446, 8916632, 896X476, 8X048X6, 8E65798, 8E794X2, 8E93156, 9015EXX, 901X2E2, 90E1114, 912X406, 9163508, 9276358, 930E6EX, 9346772, 9359842, 939EX82, 93E6768, 9411850, 9459172, 94711X8, 9479X76, 9564044, 958E6E8, 9675326, 9676778, 96XEX42, 9773346, 97X6712, 9805E34, 997X728, 99X0X5X, 9X10X46, 9X18138, 9X271EX, 9X28482, 9X9EXE8, 9XX0E60, 9E43530, 9E4XE7X, 9E50892, 9E5E798, 9E8407X, X016010, X230014, X288X06, X2X4840, X40286X, X4E8X62, X502698, X625582, X68E822, X803038, X931510, XX8148X, XXXX49X, E02X322, E056E58, E068604, E0E4528, E30XEX6, E348814, E37001X, E37025X, E4206X2, E42X5E8, E45X994, E477802, E4X2602, E500068, E565E22, E5820X4, E587856, E61840X, E681312, E743924, E777956, E83E6E4, E9X7216, E9E2670, EX65898, EX760XX, EE9E404, EEE2040, ... 1361364×139520E+1
76171X×139013X+1
2E4E10×13885E1+1
16E39X0×134070X+1
1194764×1339783+1
1481556×1335933+1
14 32759 {7, 11, 15, 181} All k=4*q^4 for all n:
let k=4*q^4
and let m=q*2^n; factors to:
(2*m^2 + 2m + 1) *
(2*m^2 - 2m + 1)
k = = 2 mod 3 (3)
k = = 4 mod 5 (5)
12 1824, 3X33, 5X87, 9XX6, 10311, 12286, 15711, 1888X, 1E476, 208X1, 21597, 295X3, 2X91X, 31490 E9EX×142739X41+1
2800X×14926382+1
28XXX×147X4X66+1
17591×14709300+1
104X4×1450E321+1
116X3×14390EEE+1
k = 1544 and 1E194 proven composite by full algebraic factors.
k = 31E14 is GFN with no known prime.
15 1E2 {3, 5, 25} k = = 1 mod 2 (2) 1 184 19X×1590100+1
114×1580114+1
78×152583E+1
74×152998+1
X×15950+1
11X×15750+1
16 292 {5, 11, 17} k = = 14 mod 15 (15) 0 (proven) none (proven) X2×161211EX+1
279×1611E50+1
203×161493+1
31×16321+1
262×16196+1
X3×16178+1
k = 16 and 230 are GFN with no known prime.
17 30X986 {5, 7, 11, X7, 541} k = = 1 mod 2 (2)
k = = 2 mod 3 (3)
395 X06, 1566, 16X4, 2196, 2736, 2806, 3E24, 62X0, 8510, 8XXX, 8XE0, 9054, 949X, X224, E006, E62X, E91X, E950, 10680, 123XX, 128XX, 1526X, 15374, 15404, 15584, 19186, 1X244, 1X786, 1X890, 20X1X, 21556, 21636, 22200, 22646, 2287X, 25010, 2681X, 28164, 29374, 2X326, 2E28X, 2E740, 30X9X, 3134X, 314EX, 31526, 31710, 32464, 3297X, 33520, 33EE6, 35X80, 36146, 36880, 38206, 39004, 3X08X, 3X960, 3X9XX, 3E3E6, 4034X, 40874, 40930, 40970, 41720, 4203X, 42176, 42976, 42E0X, 44114, 44120, 45334, 47080, 48080, 4887X, 49444, 4EX94, 500X6, 502X6, 5045X, 51654, 523X6, 53140, 5406X, 54254, 5611X, 56330, 5805X, 58760, 59560, 597X6, 59894, 5X86X, 5XE00, 5EE36, 618XX, 61904, 6337X, 63434, 63E50, 6424X, 643X6, 64830, 64EE0, 678XX, 6X050, 6X630, 6X680, 6X75X, 6E34X, 6E560, 71504, 72476, 73620, 758EX, 77290, 7821X, 782E0, 794E4, 79506, 7X296, 7X414, 805E6, 82526, 83464, 8364X, 8385X, 8436X, 856EX, 86154, 8663X, 866X4, 87914, 88190, 900EX, 91594, 93436, 93694, 93790, 95690, 96336, 97684, 99106, 9939X, 9XE80, 9E5EX, X3190, X3576, X4344, X4844, X77E0, X8260, X9326, XX186, XE074, XE45X, E012X, E0516, E0866, E0X90, E168X, E2004, E2X2X, E5830, E5936, E6560, E6850, E75X4, E86EX, E9E20, E9E36, EX286, EX324, EE22X, 10014X, 100264, 100310, 100504, 100EXX, 101100, 1011E0, 101346, 101810, 101926, 101940, 102X24, 103306, 103866, 104286, 105344, 105650, 107276, 107436, 10800X, 108164, 108550, 10E634, 10EE5X, 111664, 111786, 113134, 113X20, 113E50, 114110, 114150, 11432X, 115E04, 117906, 11823X, 11E034, 11E340, 120E3X, 122756, 122X2X, 123290, 12332X, 123814, 123830, 12548X, 1289EX, 12X68X, 12E5EX, 12E826, 12E850, 130X56, 130X8X, 131984, 131X9X, 133026, 133980, 134054, 134150, 134X1X, 13544X, 135836, 136040, 138674, 139190, 13XX30, 1402XX, 141324, 14171X, 14409X, 14584X, 145880, 145X84, 146116, 146E8X, 14702X, 14844X, 14E26X, 14E614, 150586, 150X2X, 152146, 152X66, 153084, 1548E6, 156134, 157636, 157646, 15813X, 158330, 1595EX, 1596X6, 159956, 15X174, 15X3EX, 15E846, 1602XX, 1613E6, 161726, 1622EX, 16234X, 164626, 165024, 1661X6, 166404, 166590, 166910, 169896, 16E9EX, 17165X, 171XX4, 171E84, 172310, 172716, 173226, 173266, 1752EX, 175326, 175EE6, 1769E6, 17X084, 17X594, 17E044, 17E73X, 181E60, 182200, 182486, 1832E0, 183E80, 184094, 1881E6, 188930, 18E864, 190E7X, 191906, 191XEX, 191EE4, 193424, 193X30, 1941E0, 1941EX, 194E70, 195X74, 196140, 198144, 19829X, 19865X, 1990E4, 19E210, 19E22X, 19E866, 1X05E0, 1X0690, 1X32X0, 1X4656, 1X4770, 1X5166, 1X725X, 1X7516, 1X921X, 1X972X, 1X9X6X, 1XX98X, 1XE95X, 1E0080, 1E0356, 1E0580, 1E0E2X, 1E1540, 1E221X, 1E3714, 1E457X, 1E5790, 1E57XX, 1E6614, 1E6776, 1E8206, 1E8300, 1E848X, 1EX590, 1EE04X, 1EE294, 1EE654, 201394, 201690, 202206, 2051E4, 20598X, 20602X, 206E0X, 209566, 2098E6, 20X4X6, 211006, 2115X6, 212X5X, 213326, 21566X, 218EE6, 21906X, 21X080, 21X274, 21XE6X, 2208E6, 222674, 223984, 223X8X, 227384, 2280E4, 229346, 229E96, 22E9E4, 230490, 231056, 23199X, 232644, 232XE6, 2330XX, 233164, 234010, 234220, 234236, 234796, 234X6X, 235344, 236214, 236476, 236596, 236780, 23679X, 237104, 237410, 237E54, 239114, 239710, 24226X, 24356X, 244634, 244E76, 244E94, 245226, 24583X, 245896, 246034, 246710, 247140, 247EE6, 248764, 2508XX, 251650, 252096, 252240, 252E84, 25309X, 2544X4, 254904, 254E06, 2552E0, 25539X, 2555X0, 255884, 255930, 255980, 256300, 25630X, 257184, 257476, 258820, 25E6E6, 25E730, 261276, 26146X, 261564, 261E54, 26218X, 262244, 264464, 265070, 265626, 2657X6, 2690EX, 26E196, 26E284, 26E646, 26E696, 27151X, 27228X, 273634, 274284, 2745X6, 276690, 27873X, 278766, 278EXX, 279104, 279880, 27996X, 27X180, 27X840, 27XX34, 27E420, 27E9E4, 28044X, 281110, 282370, 28320X, 283500, 283816, 283X40, 28578X, 286224, 287XE4, 28826X, 288946, 2893E6, 28X650, 28X6XX, 28X704, 28E99X, 290314, 292736, 292X6X, 29381X, 2950E6, 296016, 2976X6, 297964, 298754, 299216, 299556, 29X120, 29E0E6, 2X047X, 2X0X60, 2X12X6, 2X1956, 2X3806, 2X3976, 2X4700, 2X677X, 2X7434, 2X950X, 2XX114, 2XE86X, 2XEX10, 2E0926, 2E14EX, 2E1X4X, 2E2196, 2E2794, 2E4E90, 2E54X0, 2E633X, 2E6E20, 2E843X, 2E8964, 2E9206, 2EX5E4, 30009X, 30067X, 3006X6, 30386X, 304934, 304994, 305180, 305584, 306930, 307094, 30736X, 3076X0, 309136, 30X84X, 30X92X 29911X×178X542+1
211854×178X017+1
216530×1789739+1
149200×1789306+1
2E755X×1788E58+1
178X40×17883E2+1
18 8 {3, 7} k = = 16 mod 17 (17) 0 (proven) none (proven) 6×1813+1
7×182+1
4×182+1
1×182+1[3]
5×181+1
3×181+1
19 6E6 {E, 11, 15} k = = 1 mod 2 (2)
k = = 4 mod 5 (5)
0 (proven) none (proven) 9X×19E5X1+1
64X×19172+1
514×1915E+1
694×1970+1
696×1937+1
406×1933+1
1X 3X5X {5, 1E, 81} k = = 2 mod 3 (3)
k = = 6 mod 7 (7)
1 2E74 E23×1X2E77X4+1
1130×1X151755+1
2549×1X127E52+1
3337×1X486X6+1
38X6×1X22403+1
2E19×1X11E00+1
k = 1X and 344 are GFN with no known prime.
1E 132 {3, 5, 45} k = = 1 mod 2 (2)
k = = X mod E (E)
0 (proven) none (proven) 58×1E157447+1
8×1E58EX7+1
X2×1E8169+1
X4×1E197X+1
10X×1E1816+1
68×1E3EE+1
20 158X3 {5, 7, 11, 61, 67} k = = 1X mod 1E (1E) 51 468, 777, 10X3, 10E4, 1304, 143E, 15E6, 1E78, 205X, 2109, 27EX, 29EX, 2E75, 30E0, 30E8, 314X, 34X9, 355X, 4264, 4596, 4658, 47E2, 4823, 4881, 5453, 5474, 5745, 5761, 5976, 5X64, 6117, 6696, 7609, 7627, 7E30, 9269, X046, X38E, X418, X506, X650, XE3E, E01E, E072, E84E, EEE7, 10503, 10X95, 10E30, 11013, 11970, 11996, 12414, 13038, 1325E, 139X8, 14X03, 14E99, 15074, 1513X, 15169 8114×20171X8E+1
2286×20165E46+1
11X65×20154452+1
4X15×20153E77+1
218X×20150984+1
74X7×201503E7+1

Riesel problem base b[]

Links[]

[8]

[9]

[10]

k = 2 through 10 for bases up to 720

k = b−1 and b+1 for bases up to 720

k = 2 for bases up to 1000

k = b−1 for bases up to 1000

Table for bases up to 20[]

All n must be ≥ 1.

k-values with at least one of the following conditions are excluded from the conjectures:

1. All n-values have a single trivial factor.

2. Make a full covering set with all or partial algebraic factors.

k-values that are a multiple of base (b) and where k−1 is composite are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

base conjectured smallest Riesel number covering set ks that make a full covering set with all or partial algebraic factors trivial ks number of remaining k remaining k top 6 primes comments
2 206817 {3, 5, 7, 11, 15, 181} (none) 43 13E1, 5405, 11845, 1652E, 1X321, 23007, 32X11, 3728E, 3XX95, 4637E, 4826E, 52157, 5X655, 627X7, 6X947, 70995, 79685, 9380E, 9E29E, 9E41E, X4817, XXX61, E3747, EX135, 100837, 1329E5, 134X9E, 13760E, 138197, 13975E, 142E1E, 1464X7, 14761E, 14E252, 155X15, 156327, 157207, 15739E, 15924E, 15E271, 16664E, 167X15, 171917, 179161, 195391, 1X8111, 1XX70E, 1E0467, 1E08E2, 1E4EE1, 1EX387 112555×22EX9280−1
202X11×224X42XE−1
174E4E×2249E080−1
1E5E1×22344139−1
128067×222847X5−1
172407×221961X3−1
3 1028027E722 {5, 7, 11, 15, 17, 31, 35, 141, 531} k = = 1 mod 2 (2) 73008 129449X, 2378918, 3607022, 36E7288, 3743E2X, 5799174, 60X83X8, 71X60E2, 9913X78, X4X12X2, 10X3718X, 115557X4, 11641842, 1537XX92, 15532614, 16X0497X, 17950744, 17X1699X, 1898066X, 18X9447X, 19X6E322, 1EX3E814, 2561E3XX, 27673X84, 28839858, 2X64145X, 2X808334, 300E6E12, 31229342, 3267E738, 344567X8, 3493X544, 3562727X, 365675X4, 36E05E42, 38802584, 388X0244, 3964849X, 3X5794E8, 45623128, 45720874, 47733170, 4EE18XE2, 5049601X, 52850E42, 5618103X, 5806E38X, 58998584, 5X22EE42, 5E02X43X, 5E820694, 62498228, 633040E4, 64301548, 65951X5X, 66465312, 67X04768, 68215612, 68360328, 6886X732, 696E8E14, 69951564, 69E9X428, 6X38E168, 6E235E0X, 701X8282, 710486XX, 719X6362, 71E37988, 762EX038, 76494672, 7697E61X, 76XEEXE8, 783E3X4X, 78E29104, 7931E782, 80728XX4, 80926704, 841346E8, 855320E8, 8654E44X, 8740X544, 8761X204, 88002982, 8X0441X8, 8X77E302, 91457178, 91X4985X, 944E5862, 94853X30, 9485571X, 96503274, 96910634, 9855745X, 9X47286X, X0573678, X072942X, X130X44X, X132X9X4, X19170X4, X22005E4, X2639998, X30E803X, X9E6067X, XX3091X4, XX93239X, XE8X09X2, E5904524, E7526034, EX599EE2, ..., 1027421297X, 102746E7262, 10275802472, 1027608295X, 1027620E62X, 10276395E2X, 10276679634, 1027879E248, 1027899X56X, 1027E3X3762, 1027E936644, 1027E99575X 63005504×334E917−1
359134X34×334E809−1
1XE26EE14×334E318−1
10163226X×334976X−1
2E8X6222X×3349316−1
173838204×3347XX0−1
4 1E143 {5, 7, 11, 17, 61, 91} All k = m^2 for all n; factors to:
(m*2^n - 1) *
(m*2^n + 1)
k = = 1 mod 3 (3) 7 27X2, 5405, 5613, 8389, E320, 11845, 1652E E289×46250X1−1
E729×4348262−1
46E×417376X−1
8026×415E164−1
9826×476930−1
1X885×46E084−1
k = 3^2, 6^2, 9^2, 10^2 (etc. repeating every 3m) proven composite by full algebraic factors.
5 148842 {3, 7, 11, 27, 421} k = = 1 mod 2 (2) 53 211X, 2X0X, 11X02, 13212, 18494, 26762, 30E3X, 31472, 33518, 3520X, 3822X, 3XE52, 45948, 4E602, 4E6E4, 53272, 53692, 536E2, 6171X, 64374, 6584X, 67204, 6E154, 7021X, 70238, 70E18, 71684, 734XX, 74264, 75738, 782X4, 7E498, 8272X, 82XX4, 86X3X, 8967X, 908E8, 9199X, 92192, 95428, 9892X, 9X34X, 9E892, X3X04, E20X2, E641X, E6612, EX5EX, EEX02, 105E14, 10991X, 10X82X, 110E0X, 112452, 11933X, 119428, 124862, 127E18, 13125X, 134232, 138742, 142132, 14412X X00E2×510162X6−1
E6172×5EE8252−1
70788×5EX50XX−1
18888×5EX0552−1
13676X×5E30X17−1
43224×5E2EE45−1
6 41013 {7, 11, 27, 31, 81} k = = 1 mod 5 (5) 1 E11 19344×66E1333−1
21562×62356X2−1
38EX7×6230609−1
25635×621602E−1
28EEE×61E3X09−1
3983E×61X1142−1
7 670E5X1X7X2 {5, 11, 17, 37, 61, 131, 141, 841} k = = 1 mod 2 (2)
k = = 1 mod 3 (3)
(not full tested) 1328X0, 554896, 661746, 851430, X03428, X25X22, X6XE60, 1191180, 12234E6, 1250398, 1374500, 1545782, 1547X40, 1565162, 1574548, 1676506, 1726256, 176X746, 1808230, 186X842, 19523E2, 1X27X48, 1X83496, 2129882, 21E8218, 2237486, 2262148, 2430520, 2471X16, 2550566, 25E1548, 2643362, 2846E86, 296E072, 2987030, 2X88150, 2E12418, 2E147X6, 30279X6, 3328998, 3360648, 3380598, 346X978, 34701E8, 351EXX8, 3542466, 35935E6, 3621462, 3809E82, 3926X08, 39681E6, 3985186, 3X38392, 3XE6162, 3EEEE16, 4005220, 401XE82, 4036796, 405X548, 40E8218, 4138700, 4149328, 4354632, 4501638, 4538420, 4564EE8, 4696942, 4820262, 4863X50, 49E0400, 4XXE626, 4E66622, 4E9X972, 50X0460, 5170890, 52X71E2, 5321022, 5357156, 54E6418, 5574970, 55E1X08, 5784E80, 5794E28, 5816928, 5987940, 5E368E2, 5E74752, 6102150, 6174696, 6315876, 635E862, 6443XE0, 6574918, 6655158, 675EE28, 67EX958, 6X60120, 6E72E92, 7045748, 7083506, 7094536, 70X7710, 7216X40, 7480192, 7616XX0, 765X5E0, 77224X0, 77371X2, 7740842, 7771858, 7821E60, 7X9X992, 7E86572, 800XE92, 802XX16, 80XX352, 815X220, 8245528, 82X4362, 82E8056, 8504230, 8634150, 8654388, 8702E16, 8704X48, 8717992, 897E3X2, 8X960X2, 91E37X8, 92EE210, 9362898, 938E482, 9401X28, 96315X2, 96X1688, 9707246, 97115E0, 9771386, 9881400, 9903E02, 9977026, 9E60298, 9E71292, X236538, X2E88X2, X354410, X379952, X38X690, X666576, X6817E8, X823270, X8403E0, X8950X6, XXE7552, XE25540, E1305E8, E169398, E195756, E224X46, E382090, E437346, E44E932, E6613E2, E669620, E69X162, E794410, EX1E320, EX765X2, EX83488, EXX7586, EXX9156, EE22286, EEX5616, ... 139E42×7124717−1
260828×7E6E10−1
55X4X0×7E2392−1
345666×792322−1
9915166×772415−1
10662888×772400−1
8 12 {3, 5, 11} k = = 1 mod 7 (7) 0 (proven) none (proven) E×816−1
5×84−1
10×83−1
7×83−1
2×82−1
11×81−1
9 62 {5, 7, 11, 61} All k = m^2 for all n; factors to:
(m*3^n - 1) *
(m*3^n + 1)
k = = 1 mod 2 (2) 0 (proven) none (proven) 20×98−1
12×98−1
50×95−1
36×95−1
38×94−1
3X×93−1
k = 4, 14, 30, and 54 proven composite by full algebraic factors.
X 5X80 {7, E, 11, 31} k = = 1 mod 3 (3) 1 2685 408E×X366025−1
4E6E×X160010−1
3X35×X2XX48−1
1153×X25EE8−1
1063×X22676−1
93E×X1523E−1
E 5EX {3, 7, 17, 31} k = = 1 mod 2 (2)
k = = 1 mod 5 (5)
0 (proven) none (proven) 52×E131E6−1
218×E310−1
124×E137−1
1E8×E136−1
372×E66−1
328×E66−1
10 274 {5, 11, 25} (Condition 1):
All k where k = m^2
and m = = 5 or 8 mod 11:
for even n let k = m^2
and let n = 2*q; factors to:
(m*10^q - 1) *
(m*10^q + 1)
odd n:
factor of 11
(Condition 2):
All k where k = 3*m^2
and m = = 3 or X mod 11:
even n:
factor of 11
for odd n let k = 3*m^2
and let n=2*q-1; factors to:
[m*2^n*3^q - 1] *
[m*2^n*3^q + 1]
k = = 1 mod E (E) 0 (proven) none (proven) 111×101E9−1
3X×10142−1
214×1034−1
197×1034−1
7X×1030−1
204×1026−1
k = 21, 54, and 230 proven composite by condition 1.
k = 23 and 210 proven composite by condition 2.
11 212 {5, 7, 15} k = = 1 mod 2 (2)
k = = 1 mod 3 (3)
0 (proven) none (proven) 200×1153255−1
102×1126−1
78×111E−1
86×1118−1
210×11X−1
160×11X−1
12 4 {3, 5} k = = 1 mod 11 (11) 0 (proven) none (proven) 2×124−1
3×121−1
13 40E498E248436 {11, 15, 95, 157, 181, X41, 22E1} k = = 1 mod 2 (2)
k = = 1 mod 7 (7)
(not full tested) 164X96, 1506E08, 1611X60, 1906942, 251E298, 2X30E62, 3071432, 3326098, 3457132, 35294X4, 3532268, 353XX60, 36187X2, 361XE52, 3665894, 36XE112, 372E10X, 3779834, 388241X, 38XE852, 3980114, 3989556, 3X67E6X, 3X7099X, 3XXE808, 3E62E18, 3E9805X, 3EXEEE2, 3EE54E2, 4029152, 4101770, 410213X, 411078X, 411X984, 41454E8, 4212502, 42687X2, 42X8556, 4308326, 4335436, 4340878, 4352722, 4432X26, 4513516, 452325X, 45E5X70, 490242X, 491733X, 4942698, 4970446, 4X25EX8, 4E4954X, 4E4X088, 500001X, 5102206, 514E280, 51X4764, 5226226, 52650X8, 5309008, 5331E04, 53E5E2X, 5449410, 54XE06X, 552E994, 557E3X6, 5711830, 57E8E38, 5818444, 583E778, 5969394, 5X13666, 5X17E80, 5X21874, 5X38126, 5X664X2, 5E30E68, 5E44604, 602702X, 612EE12, 614E450, 6183592, 622XX80, 6235X54, 6239358, 62775X8, 6323014, 639629X, 63E0994, 64EE640, 669E894, 6728X6X, 6827384, 6839426, ... 1E63996×13188060−1
323X126×131614X3−1
1155348×13148X59−1
137670X×131135XE−1
1915466×13105017−1
1883X06×13100545−1
14 177X5 {7, 11, 15, 181} All k = m^2 for all n; factors to:
(m*4^n - 1) *
(m*4^n + 1)
k = = 1 mod 3 (3)
k = = 1 mod 5 (5)
14 30E, 13E5, 5613, 78E0, 7E1E, X748, E320, E538, 10583, 11845, 12563, 13322, 1491E, 15365, 1652E, 1713E 734E×14258327−1
2118×14190042−1
E729×14184131−1
4535×14157E47−1
1771E×14E49E0−1
3E70×14X53E7−1
k = 3^2, 10^2, 13^2, 16^2, 23^2, 26^2, (etc. pattern repeating every 26m) proven composite by full algebraic factors.
15 72 {3, 5, 25} k = = 1 mod 2 (2) 0 (proven) none (proven) 38×153908−1
30×15183−1
X×1599−1
22×1592−1
4X×152E−1
3X×1521−1
16 186 {5, 11, 17} k = = 1 mod 15 (15) 0 (proven) none (proven) 107×162XX−1
66×16124−1
42×1692−1
67×1653−1
179×1638−1
134×1638−1
17 4600X2 {5, 7, 11, X7, 131} (Condition 1):
All k where k = m^2 and m = = 2 or 3 mod 5:
for even n let k = m^2 and let n = 2*q; factors to:
(m*17^q - 1) * (m*17^q + 1)
odd n:
factor of 5
(Condition 2):
All k where k = 17*m^2 and m = = 2 or 3 mod 5:
[Reverse condition 1]
k = = 1 mod 2 (2)
k = = 1 mod 3 (3)
670 266, 2332, 4772, 6X80, 7996, 9780, 9E18, X418, 101X6, 102E8, 10E30, 11518, 12086, 12088, 12168, 13E60, 14812, 15760, 15E46, 17906, 17X86, 1E438, 1E480, 20730, 20818, 20X02, 21416, 22848, 23356, 24922, 24996, 249E2, 25012, 256X8, 25E66, 26900, 270X6, 28068, 28452, 29738, 2E408, 2EX30, 30120, 30646, 31130, 313X8, 325X0, 33168, 33826, 35558, 35898, 36242, 364E2, 36688, 36786, 36898, 37122, 37316, 387X6, 39432, 3X726, 3E9E0, 40858, 40E98, 41020, 42048, 42522, 42756, 43152, 43250, 44398, 45892, 46922, 47602, 47758, 4X6X8, 4XE98, 4E212, 4E440, 51170, 51338, 51820, 51936, 52146, 52XE6, 53882, 54262, 54596, 55E42, 56292, 562E8, 56556, 565X8, 56620, 56E70, 58XX2, 59560, 5X232, 5E196, 5E8X6, 61148, 621X8, 62482, 63126, 641E2, 64418, 65202, 66130, 66288, 67336, 6X688, 6X702, 6XE56, 6E318, 71520, 72008, 73XX0, 74448, 74672, 75302, 75836, 767X6, 76978, 781E2, 78832, 79260, 79502, 7X096, 7X192, 7X200, 80E36, 81E22, 82X42, 83088, 838X2, 86550, 86698, 87550, 87858, 88046, 89038, 8E402, 8E526, 904E2, 91192, 92802, 93152, 934E8, 93638, 936X2, 93EE0, 94202, 95712, 96828, 97298, 974E8, 97556, 99816, 9X412, 9X5E0, 9E060, 9E432, 9E722, 9E802, X1250, X1888, X2750, X3332, X4352, X5080, X5312, X7306, X7308, X7500, X75X0, X7638, X9126, X9X36, XE558, E0188, E3642, E4240, E5890, E67E2, E68E8, E7490, E8810, E8878, EX092, EX8X6, EXE72, EE808, 100510, 101128, 1012E6, 101426, 1017X6, 101XE2, 103138, 103452, 103910, 103X16, 104E56, 105140, 105X20, 105E90, 106298, 106572, 1065X6, 1070X8, 107768, 108410, 108880, 1090X0, 109550, 10X770, 10E492, 10E926, 110478, 110486, 110X32, 111928, 112488, 112X52, 115250, 115EE0, 116806, 116928, 1171X6, 1175E6, 117E48, 118192, 118X60, 118E18, 11X006, 11X0X6, 11EX58, 121X38, 121XE0, 122E52, 123066, 124718, 126140, 126676, 1276E8, 129170, 129710, 129728, 129E62, 129EE2, 12X148, 12X6E6, 12E908, 12EE90, 1302X8, 130418, 130812, 131170, 131E52, 132366, 132500, 132XX0, 133190, 133228, 1335E0, 134720, 135208, 1356E8, 135X98, 136218, 137068, 137XX8, 138376, 138542, 138702, 141068, 141808, 142100, 1423E0, 142906, 1441E2, 144226, 144232, 144478, 144968, 145008, 1451X6, 145576, 147772, 148X06, 14XXX0, 14XE80, 151178, 151878, 151950, 154736, 154818, 156178, 156X22, 158138, 1582E0, 158X68, 15E070, 15E708, 15EE02, 160218, 160766, 161646, 161808, 162472, 162488, 162536, 163948, 164878, 164E26, 166460, 168932, 16X742, 16X992, 170060, 170560, 170858, 170E98, 171328, 172442, 172852, 173708, 173EE2, 174000, 175148, 175358, 176192, 1771X0, 177440, 177652, 177758, 178508, 1787X0, 1788X6, 1794E0, 179858, 180222, 180598, 180860, 181910, 182648, 182898, 1833E6, 183760, 183E80, 185022, 186602, 187576, 187906, 188EX8, 18XX30, 18XX46, 18XE00, 18E726, 190106, 191866, 192432, 192666, 1943X6, 194572, 194700, 195042, 195648, 195728, 195E46, 196116, 196158, 196390, 196542, 197722, 1988E0, 198966, 198E60, 199152, 199XX8, 19X016, 19X568, 19E172, 19E710, 1X0118, 1X0682, 1X0692, 1X3058, 1X30E6, 1X4488, 1X4956, 1X50E2, 1X5860, 1X5876, 1X5EX8, 1X6186, 1X6570, 1X6E18, 1X7102, 1X7450, 1X8026, 1X8630, 1X9690, 1X9788, 1XX140, 1XEEE0, 1E1946, 1E3386, 1E3836, 1E4110, 1E4716, 1E5530, 1E5X48, 1E6988, 1E8060, 1E8212, 1E8690, 1E8852, 1E8X46, 1E9506, 201X12, 202730, 203870, 2038E2, 2043E6, 204566, 205536, 205728, 207X06, 207E68, 209120, 2098X8, 20XEX8, 20E770, 20E852, 210716, 211292, 211348, 211768, 212122, 2132X2, 213930, 213E16, 214480, 214X20, 214E90, 215298, 215802, 216396, 2179E2, 217E88, 219392, 21X0X6, 21X222, 21X932, 21X976, 21E196, 2202E0, 2212E0, 222E66, 222EE6, 2249E0, 226438, 226E08, 226E16, 226E32, 227466, 227E42, 228E52, 229170, 22XX92, 22E092, 230886, 231X30, 232346, 232662, 233236, 234552, 235220, 235272, 235416, 236968, 236E50, 237258, 237372, 237766, 238470, 239390, 239E56, 23XX02, 23EEX0, 240320, 240322, 240960, 241486, 241X18, 242368, 242E58, 244780, 244976, 2450X6, 246382, 247560, 248108, 248720, 24E910, 250018, 251078, 251406, 2521E8, 252440, 252618, 253136, 2538E2, 2539E0, 255E80, 256966, 257352, 258428, 258468, 2588X6, 259162, 25E246, 25E818, 25E972, 260676, 260X20, 2617E8, 261950, 264020, 264756, 264860, 264922, 265312, 265838, 265966, 2664E8, 266830, 267170, 2678E6, 268XX0, 26X360, 26X602, 26XE18, 26E5E6, 2708X8, 271172, 271358, 271578, 271962, 271E58, 272552, 272802, 273916, 277088, 2772E2, 277430, 277546, 277836, 277932, 278062, 279102, 279678, 279898, 279992, 27X486, 27XX08, 27XEX6, 27E720, 27EX02, 280270, 281838, 281940, 282700, 283778, 284190, 285140, 286096, 286766, 286X06, 287822, 28X052, 28E458, 28EE92, 291E46, 293182, 293406, 293560, 293562, 294780, 294992, 295058, 296236, 298640, 29X040, 29X098, 29E612, 29EXE6, 2X04E2, 2X3200, 2X5246, 2X5858, 2X63E0, 2X6448, 2X65E0, 2X8688, 2X97X2, 2X99X2, 2XX270, 2XX288, 2XX682, 2XE078, 2XE5E2, 2E0328, 2E0378, 2E0932, 2E1080, 2E1556, 2E1718, 2E25X8, 2E3826, 2E3X26, 2E4756, 2E6016, 2E6398, 2E65E6, 2E69E0, 2E7202, 2E7750, 2E7X92, 2E8180, 2E8868, 2E8980, 2E9608, 2E9930, 2E9X38, 2EE992, 300622, 301510, 302002, 303232, 303692, 303E26, 3059E0, 306210, 306866, 307EX0, 30X058, 30XE22, 30E738, 312322, 312768, 3131X6, 315070, 315762, 315XE2, 316X48, 317136, 317862, 317XX6, 319368, 3193X8, 31X360, 31E006, 31E150, 31EX30, 320008, 320056, 320362, 321526, 322156, 3225X0, 3225X2, 323X36, 324X22, 324X38, 3254X0, 326172, 326356, 327132, 3273X0, 327622, 328XE0, 329442, 32X506, 32X902, 32E0X8, 32E778, 32E786, 32E830, 330X86, 3321X0, 332432, 333396, 333X90, 334E96, 335866, 336756, 338072, 338718, 33E376, 343042, 3433E8, 343X36, 344578, 3453E0, 346812, 346E02, 347252, 347988, 348300, 348762, 348E48, 349296, 349826, 349876, 34X142, 34X618, 34XX62, 34E216, 34E928, 34EX66, 34EE38, 350196, 350526, 351340, 3513X0, 352052, 3525E2, 353X30, 356092, 357342, 357498, 358048, 358126, 3592E6, 35X566, 360666, 360XX2, 361E60, 3638X6, 364326, 3643E6, 3653E0, 365432, 365730, 366230, 366X98, 369560, 36X038, 36E730, 36EE36, 371226, 371X58, 371E56, 372140, 372972, 374088, 3769E2, 376X66, 377238, 377430, 378078, 378096, 378482, 378498, 379150, 37X402, 37X7E6, 37X950, 37EXX8, 380156, 380980, 3819X8, 382388, 383978, 384662, 385202, 385576, 385800, 385E56, 386086, 386220, 387740, 388178, 3882X0, 388610, 389372, 38X430, 38X796, 38E176, 38E9X8, 390396, 390X72, 391E28, 3923X6, 393656, 393680, 394336, 394828, 394932, 395232, 396516, 397638, 397726, 397760, 398060, 398270, 398576, 399480, 399520, 39E830, 3X0198, 3X0742, 3X0806, 3X3240, 3X3570, 3X9846, 3X9X72, 3X9E62, 3XX5E6, 3XE318, 3E0182, 3E14X0, 3E2030, 3E3170, 3E3286, 3E5230, 3E5980, 3E6838, 3E6E36, 3E7370, 3E9EX8, 400040, 400362, 401218, 401382, 402X50, 403422, 403432, 403722, 4040E2, 406670, 406960, 406E72, 407106, 407X02, 408458, 408586, 409828, 409EE0, 40X522, 40E2E8, 40E380, 410438, 410496, 4105E0, 411238, 411860, 412418, 412668, 412X62, 414E48, 4150E8, 415210, 415666, 415E22, 416X70, 416E32, 417642, 417836, 418396, 418598, 419E38, 41X7E2, 41EE16, 420012, 420386, 420660, 421008, 422E56, 422EX6, 4233X8, 4240E2, 424842, 425X88, 426516, 427202, 42X298, 42XX62, 42E0X8, 42E1E2, 4306E2, 431X10, 432238, 4323E8, 432X20, 432X60, 432EX0, 433206, 433232, 433602, 434058, 434282, 434E62, 435252, 435840, 435X16, 436068, 436752, 436X42, 437088, 4383X2, 438802, 438X06, 439478, 43E666, 4401E8, 440966, 441E50, 442790, 4430X8, 443986, 444E66, 445182, 447582, 447XE6, 448286, 44E102, 44E908, 450610, 4506E2, 450750, 451420, 4523E6, 4538X2, 4542E6, 455428, 455926, 4564X2, 457550, 458X62, 459908, 459E86, 45X1E0, 45X4X0, 45E508, 45E680 1X48×17X6026−1
1E02×175X3X0−1
374708×174998E−1
2110E6×1749891−1
237646×17497XE−1
382136×1749711−1
k = 10^2, 16^2, 24^2, 36^2, 40^2, 4X^2 (etc. pattern repeating every 26m) proven composite by condition 1.
k = 17*10^2, 17*16^2, 17*24^2, 17*36^2, 17*40^2, 17*4X^2 (etc. pattern repeating every 26m) proven composite by condition 2.
18 8 {3, 7} k = = 1 mod 17 (17) 0 (proven) none (proven) 2×18X−1
6×182−1
5×182−1
7×181−1
4×181−1
3×181−1
19 3X8 {E, 11, 15} k = = 1 mod 2 (2)
k = = 1 mod 5 (5)
0 (proven) none (proven) 54×1917XE−1
352×19696−1
10X×1987−1
70×1974−1
EX×1940−1
316×192X−1
1X 26E9 {5, 1E, 81} k = = 1 mod 3 (3)
k = = 1 mod 7 (7)
1 2148 1968×1X79344−1
2395×1X19232−1
1799×1X14233−1
705×1X13103−1
2472×1X718E−1
135×1X6749−1
1E 338 {3, 5, 45} k = = 1 mod 2 (2)
k = = 1 mod E (E)
1 298 142×1EX2230−1
E2×1E141E8−1
28X×1EE809−1
222×1E9EE0−1
328×1E4450−1
172×1E3730−1
20 16868 {5, 7, 11, 61, 401} (Condition 1):
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*20^q - 1) *
(m*20^q + 1)
odd n:
factor of 5
(Condition 2):
All k where k = 6*m^2
and m = = 1 or 4 mod 5:
even n:
factor of 5
for odd n let k = 6*m^2
and let n=2*q-1; factors to:
[m*2^n*6^q - 1] *
[m*2^n*6^q + 1]
k = = 1 mod 1E (1E) 58 285, XE9, EX7, 1612, 18X6, 220X, 2553, 3124, 3420, 3427, 3658, 3E11, 3E53, 42X1, 4629, 5XE7, 6027, 620X, 6511, 67E8, 7172, 7239, 7819, 7E0E, 8837, 8927, 9052, 948X, 9748, 9X23, X110, X546, X81X, XE84, E090, E70X, EXX8, EE17, 100EE, 101X4, 10508, 10558, 10633, 10755, 11765, 125E2, 12668, 12805, 12839, 12863, 12X83, 13194, 13219, 13519, 1386X, 14255, 14514, 14841, 1484E, 1510E, 15141, 15151, 15E93, 16262, 16446, 16526, 16624, 166XE 5X77×20106433−1
6X82×20102245−1
11417×2010216X−1
10483×20102013−1
14636×20E685X−1
10058×20E3014−1
k = 2^2, 3^2, 7^2, 8^2. 10^2, 11^2, 15^2, 16^2 (etc. pattern repeating every 5m where k not = = 1 mod 1E) proven composite by condition 1.
k = 6*1^2, 6*4^2, 6*6^2, 6*9^2, 6*E^2, 6*12^2, 6*14^2, 6*17^2 (etc. pattern repeating every Xm where k not = = 1 mod 1E) proven composite by condition 2.
  1. this prime is GFN, if GFN are not counted, then the next prime would be 39×83+1
  2. this prime is GFN
  3. this prime is GFN, if GFN are not counted, then the next prime would be 2×181+1
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