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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 (GFN), i.e. q^m*b^n+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 (Sierpinski) or k−1 (Riesel) is composite are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

Sierpinski bases[]

1 k bases[]

base conjectured k remain k
9 1434 1218
X 5387 452X
15 1E2 184
1X 3X5X 2E74
23 38X 292
37 1380 11X
58 1X 15
63 2446 914
72 24 8
73 1XX 28
94 203X E94
99 1320 X2X
9X 59 40
X2 34 2X
X8 38 34
E8 3X 8
104 2707 1449
10E 12 4
126 6 4
132 1E 8
135 28 X
148 3E 34
150 69 4
161 10X6 950
162 62 15
16E 18 16
172 8 4
190 39 23
194 859 593
195 38 34
197 100 54
19E X 8
1X0 46 35
206 9E 51
214 X1 50
225 44 38
226 100 75
228 8X 81
234 40 23
23E 8 4
244 219 147
255 14 8
271 X0 90
277 186 110
295 58 18
296 78 51
2X9 48 X
2E6 52 8
2E8 X 8
2EX 10E79 8X49
304 154 39
316 X2 73
325 8 2
32E 8 4
330 14X 81
332 112 28
338 44 24
340 32 10
350 72 59
355 14 8
368 16 5
370 EX X2
374 6EX 269
386 8X 88
39X 97 7X
3EX 20 14
403 72 6
406 46 28
418 148 E4
425 84 5X
452 5X 28
461 100 54
484 589 427
486 143 8X
492 172 98
4X6 63 33
4XE 4X 34
500 88 11
504 150 9
50X 123 70
514 E1 10
518 12 11
532 X 8
54X 154 117
551 380 26X
554 110 107
555 XX 8
559 178 80
566 137 29
56E 14 4
572 1X4 E8
580 247 109
585 E4 68
590 10 8
596 11X 75
598 28 2
5XX 269 190
5E1 150 8X
600 126 62
602 14 8
60E 62 4
617 230 56
642 4 2
645 14 2
656 18 8
65E 12 X
675 44 8
688 14 E
698 92 21
69E 34 8
6E2 32 10
6E8 4 2
701 214 100

2 k bases[]

base conjectured k remain k
22 165 55, 10E
24 2776 607, 2774
26 603 1E2, 410
41 1854 99X, X14
8E X2 32, 58
15X 123 73, 8X
178 68 45, 57
20X 133 56, 120
223 E4X 39X, 598
238 32 4, 14
23X 56 4, 43
265 214 2, 128
274 42X 186, 20X
278 X8 54, 71
28X 113 X9, E4
28E X 4, 8
2X2 E4 87, 92
301 246 9X, 1X0
310 12E 3X, 93
313 150 72, 104
327 830 434, 754
334 73 19, 57
366 162 49, 143
382 4X 12, 44
385 74 32, 54
392 28 2, 11
3E0 96 58, 67
415 X 4, 8
438 1X 10, 11
44X X6 63, 8X
490 72 2X, 35
494 89 46, 87
4X7 38X 256, 336
529 100 10, 80
53X 177 4X, 166
542 194 8, E
546 51 6, 17
55X 169 X7, 114
568 74 51, 6X
589 128 4X, 54
594 59 30, 57
595 E8 28, 8X
597 336 7X, 1E0
5E7 336 E4, 18X
61E 32 14, 2X
67X 2X4 40, 263
6X9 30 6, 8

3 k bases[]

base conjectured k remain k
31 1610 7X, 8X0, 1354
5X 64E1 32E4, 5413, 605X
84 1519 301, 64X, 12X9
86 205 X2, 12X, 178
E1 1160 74, 7XX, 934
111 940 490, 81X, 864
119 187X 884, X3X, 105X
13E 212 44, 58, 124
170 7E6 138, 233, 643
189 588 304, 490, 4XX
202 82 61, 74, 77
256 E9 10, 63, E2
268 34 8, 10, 2X
2X4 E0 30, 54, 99
2E2 3X 8, 11, 15
30E 134 8, 1X, 7X
360 149 67, 7X, 98
363 7E2 278, 4X6, 6X0
388 12X 11, 63, 69
3X6 197 8, 133, 146
3E9 150 8X, E0, 14X
402 58 15, 27, 32
45E X4 2, 62, 84
4E2 68 8, 3E, 44
4E8 17X 8X, X1, 11X
52X 211 83, 113, 147
535 X8 14, 28, 78
53E 68 4, 14, 44
558 34 12, 14, 32
577 3X0 190, 316, 38X
5X2 58 11, 17, 27
608 82 17, 3X, 58
609 154 98, 106, 152
612 1E 2, 11, 15
624 137 3X, 79, 107
635 9X 52, 58, 74
64X 1E9 26, E6, 15X
666 152 34, E5, E7
669 282 136, 208, 228
66E 68 2, 2X, 58
6X0 144 X9, 114, 142
6X1 630 37X, 476, 53X

Riesel bases[]

1 k bases[]

base conjectured k remain k
6 41013 E11
X 5X80 2685
1X 26E9 2148
1E 338 298
23 570 4XX
27 65E66 28326
79 430 2E4
7X 33 25
87 806 650
91 100 70
X3 10X 20
E1 2170 652
114 133 98
117 270 192
125 24 1X
131 210 120
159 386 206
15X 38 E
165 32 28
175 12 X
176 3X 6
191 1128 10X6
1XE 1X 4
226 100 X2
229 1X 8
232 92 42
234 40 8
238 32 16
239 35X 14
23X 56 12
24E 24 1X
256 E9 5E
266 1279 1017
268 34 30
271 62 16
288 62 7
292 8 7
295 58 32
296 78 28
2X4 59 6
2X8 E8 47
2E5 5X 54
322 14 E
331 370 2E2
332 112 E5
351 122 78
354 452 49
360 149 7X
361 4E6 58
370 EX 73
385 74 62
398 11 7
399 2X 6
39E 1X X
3XE 3X 24
3E9 150 6
405 82 2
406 46 44
42X 529 176
42E 9X X
433 2X 10
437 160 6
43X 76 66
450 23 9
462 8 4
472 12 7
475 32 8
478 12 E
487 136 126
490 3X 33
494 89 9
49E 24 22
4X6 63 28
4E4 135 3E
507 186 8
50X 123 122
50E 52 2X
514 606 3X8
51E 28 12
535 X8 28
545 38 32
552 24 12
553 212 88
554 110 98
55X 169 40
568 74 8
578 11 4
579 136 X2
57E 14 8
595 E8 88
598 28 8
5XX 269 165
600 126 96
603 78 8
613 2X 20
617 230 142
620 59 54
635 108 1X
67X 126 8
687 2X0 182
688 14 4
6XE 6X 18
70E 4 2
714 69 25
715 18 8
719 88 22

2 k bases[]

base conjectured k remain k
51 7978 56E6, 6150
94 2283 E66, 1E13
112 44 25, 38
116 1X77 1286, 1789
13E 198 150, 192
161 2868 968, 1642
162 62 45, 4E
1E5 234 32, 222
200 431 243, 365
228 8X 61, 87
241 276 32, 166
29X 415 198, 209
2X1 386 1E8, 248
304 459 116, 248
313 104 66, 9X
327 478 160, 258
355 6X 12, 48
366 162 114, 14X
384 3E9 110, 150
386 8X 25, 4E
3E0 96 33, 72
410 7X 3, 14
422 58 4X, 56
44X X6 18, 25
457 210 96, 152
477 596 200, 248
492 172 28, 78
49X 1E3 80, 1X0
4X0 200 108, 169
515 34 12, 1X
522 2X 12, 21
546 51 21, 26
550 1E9 91, X2
559 178 X0, X6
56E 58 12, 44
572 1X4 X5, 15X
57X 118 38, X8
590 100 54, 58
609 154 7X, 96
61E 32 1X, 22
657 2360 10X6, 2258
665 112 78, 94
669 282 36, 212
696 129 10, 7X
698 92 X, 27
6X4 210 3E, 79
6E7 290 190, 226
71E 38 4, 8

3 k bases[]

base conjectured k remain k
57 19X0 8X2, 1358, 1792
5X 36X8 1040, 1336, 31E8
68 191 X, 27, 15X
124 892 163, 283, 7E0
20X 98 23, 78, 89
223 630 54, 294, 3X0
227 X72 432, 950, X56
233 4X0 32, 150, 26X
26X 383 138, 169, 299
278 X8 32, 42, 67
2X2 E4 3E, 4X, 54
2E4 59 16, 19, 38
307 100 38, 80, X0
312 62 11, 47, 56
318 4E E, 37, 44
333 526 100, 212, 2X0
358 11X 32, 45, 6X
374 7X0 230, 400, 772
37X 630 X5, 1X9, 446
412 144 82, 91, 108
421 582 210, 342, 520
460 179 19, 136, 138
500 88 18, 23, 6E
580 247 16, 95, 150
624 139 40, 80, 122
625 104 1X, 38, 7X
648 87 45, 51, 64
6X0 144 83, 9E, 13E
6X3 122 4X, 7X, 9X
6E5 814 5X, 122, 6XX
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