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1st Page 2007

Mersenne Thread

k < 300

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10⁶ < k < 10⁷

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10¹⁰ < k < Infin.

Condensed
10000 < k < 100000

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Constant-n search

k < 300

300 < k < 2000

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Status 300 <k< 1200

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k = 5 - 62.04%

k = 15 - 58.20%

k = 17 - 67.80%

k = 65 - Done

k = 105 - 48.38%

k = 125 - Done

9 k's - Done

...

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Drive #14 - .%

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Biggest Puzzles

The Riesel-Problem for Twins and Sophie Germains

Started: 2008-11-09 Last update: 2020-05-17

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