The Riesel-Problem for Twins and Sophie Germains
Special thanks to W.Keller for his data and corrections.
Definition :
Twins and Sophie Germains for k · 2n - 1 can only be produced by a k divisible by 3.
So there follows the question:
Exist a k (k > 1, divisible by 3) which never produce a Twin (k · 2n - 1 and k · 2n + 1 are prime)
or Sophie Germain (k · 2n - 1 and k · 2n+1-1 are prime) for any n > 0?
Conjecture 1: k = 237 is the smallest k never producing a Twin prime pair.
Conjecture 2: k = 807 is the smallest k never producing a Sophie Germain pair.
References:
Problem 49 from 'The Prime Puzzles & Problem Connection': Sierpinski-like numbers
Data :
a) no Twin pair:
W.Keller, C.Nash and T.Masser independently found this:
k = 237 is the smallest value never producing a Twin prime pair with covering set = [5, 7, 13, 17, 241].
(Others: k = 807 with cs = [5, 7, 13, 19, 37, 73] and k = 4581 with cs = [5, 7, 13, 17, 241])
There are 9 values k < 237 without Twin prime pair found so far:
111, 123, 153, 159, 171, 183, 189, 219, 225.
T.Masser searched these 9 k upto n = 700000.
The search was done up to n = 1,000,000 by RPS Team Drive 9k's.
b) no Sophie Germain pair:
W.Keller found this:
k = 807 is the smallest value never producing a Sophie Germain pair with covering set = [5, 7, 13, 19, 37, 73].
There are 32 values k < 807 without known Sophie Germain pair:
39, 183, 213, 219, 273, 279, 333, 351, 387, 393, 399, 417, 429, 471, 531, 543,
561, 567, 573, 591, 597, 603, 639, 681, 687, 693, 699, 723, 753, 759, 771, 795.
c) all Overview:
k divisible by 3 with their first Twin and Sophie Germain (n given) and search ranges.
Colors: Done / Twin not needed k > 237, but listed too / Reserved by person / Reserved by project
| k | Twin | SG | max n | note |
|---|
| 3 | 1 | 1 | - | |
| 9 | 1 | never | - | no SG (k square) |
| 15 | 1 | 1 | - | |
| 21 | 1 | 1 | - | |
| 27 | 2 | 1 | - | |
| 33 | 6 | 2 | - | |
| 39 | 3 | - | 4.848M | M.Kwok |
| 45 | 2 | 1 | - | |
| 51 | 1 | 9 | - | |
| 57 | 2 | 1 | - | |
| 63 | 14 | 2 | - | |
| 69 | 1 | 4 | - | |
| 75 | 1 | 5 | - | |
| 81 | 5 | never | - | no SG (k square) |
| 87 | 2 | 1 | - | |
| 93 | 4 | 3 | - | |
| 99 | 1 | 4 | - | |
| 105 | 2 | 2 | - | |
| 111 | - | 2 | 3.0M | RPS |
| 117 | 4 | 1 | - | |
| 123 | - | 2 | 2.02M | RPS |
| 129 | 3 | 3 | - | |
| 135 | 1 | 9 | - | |
| 141 | 1 | 1 | - | |
| 147 | 44 | 1 | - | |
| 153 | - | 3 | 2.02M | RPS |
| 159 | - | 4 | 2.02M | RPS |
| 165 | 2 | 2 | - | |
| 171 | - | 2 | 2.02M | RPS |
| 177 | 12 | 12 | - | |
| 183 | - | - | 2.02M | RPS |
| 189 | - | 3 | 2.02M | RPS |
| 195 | 4 | 3 | - | |
| 201 | 3 | 9 | - | |
| 207 | 2 | 20 | - | |
| 213 | 36 | - | 1.65M | K.Wozny |
| 219 | - | - | 2.102M | RPS |
| 225 | - | never | 2.0M | no SG (k square) |
| 231 | 1 | 6 | - | |
| 237 | never | 17 | - | no Twin possible |
| 243 | 12 | 11 | - | |
| 249 | - | 8 | 2.0M | |
| 255 | 2 | 1 | - | |
| 261 | 1 | 114 | - | |
| 267 | 4 | 4 | - | |
| 273 | 2 | - | 2.0M | |
| 279 | - | - | 2.0M | |
| 285 | 1 | 7 | - | |
| 291 | 1553 | 29 | - | |
| 297 | 14 | 1 | - | |
| 303 | - | 6 | 2.71M | NPLB |
| 309 | 1 | 4 | - | |
| 315 | 22 | 4 | - | |
| 321 | 1 | 1 | - | |
| 327 | 4 | 1 | - | |
| 333 | 54 | - | 2.71M | NPLB |
| 339 | 3 | 7 | - | |
| 345 | 4 | 6 | - | |
| 351 | - | - | 2.0M | |
| 357 | 2 | 4 | - | |
| 363 | 2 | 2 | - | |
| 369 | 13 | 4 | - | |
| 375 | 3 | 2 | - | |
| 381 | 17 | 1 | - | |
| 387 | 28 | - | 2.0M | |
| 393 | - | - | 2.0M | |
| 399 | 11 | - | 2.0M | |
| 405 | 1 | 1 | - | |
| 411 | 1 | 13 | - | |
| 417 | 2 | - | 1.59M | NPLB |
| 423 | 8 | 22 | - | |
| 429 | 1 | - | 1.59M | NPLB |
| 435 | 4 | 13 | - | |
| 441 | 1 | never | - | no SG (k square) |
| 447 | 2 | 4 | - | |
| 453 | 48 | 2 | - | |
| 459 | 3 | 8 | - | |
| 465 | 6 | 5 | - | |
| 471 | 3 | - | 1.59M | NPLB |
| 477 | - | 1 | 1.59M | NPLB |
| 483 | 2 | 2 | - | |
| 489 | 5 | 3 | - | |
| 495 | 16 | 8 | - | |
| 501 | - | 2 | 1.59M | NPLB |
| 507 | 2 | 1 | - | |
| 513 | 6 | 267 | - | |
| 519 | 11 | 7 | - | |
| 525 | 1 | 1 | - | |
| 531 | 1 | - | 540k | NPLB |
| 537 | 6 | 5 | - | |
| 543 | - | - | 540k | NPLB |
| 549 | 11 | 3 | - | |
| 555 | 9 | 30 | - | |
| 561 | 7 | - | 540k | NPLB |
| 567 | 2 | - | 540k | NPLB |
| 573 | 344 | - | 540k | NPLB |
| 579 | - | 35 | 540k | NPLB |
| 585 | 2 | 2 | - | |
| 591 | 5 | - | 540k | NPLB |
| 597 | 70 | - | 540k | NPLB |
| 603 | 10 | - | 540k | NPLB |
| 609 | 19 | 3 | - | |
| 615 | 1 | 1 | - | |
| 621 | 3 | 22 | - | |
| 627 | 6 | 5 | - | |
| 633 | 32 | 42 | - | |
| 639 | 1 | - | 540k | NPLB |
| 645 | 1 | 1 | - | |
| 651 | 1 | 13 | - | |
| 657 | 58 | 29 | - | |
| 663 | 24 | 3 | - | |
| 669 | 11 | 8 | - | |
| 675 | 5 | 2 | - | |
| 681 | 31 | - | 540k | NPLB |
| 687 | 34 | - | 540k | NPLB |
| 693 | - | - | 540k | NPLB |
| 699 | 5 | - | 540k | NPLB |
| 705 | 3 | 1 | - | |
| 711 | 17 | 5 | - | |
| 717 | 12 | 4 | - | |
| 723 | 6 | - | 540k | NPLB |
| 729 | - | never | 540k | no SG (k square), NPLB |
| 735 | 3 | 2 | - | |
| 741 | 1 | 1 | - | |
| 747 | 6 | 8 | - | |
| 753 | - | - | 540k | NPLB |
| 759 | 17 | - | 540k | NPLB |
| 765 | 4 | 9 | - | |
| 771 | - | - | 540k | NPLB |
| 777 | 10 | 20 | - | |
| 783 | - | 3 | 540k | NPLB |
| 789 | - | 23 | 540k | NPLB |
| 795 | 3 | - | 540k | NPLB |
| 801 | - | 1 | 540k | NPLB |
| 807 | never | never | 540k | no Twin, no SG possible |