# 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

kTwinSGmax nnote
311-
91never-no SG (k square)
1511-
2111-
2721-
3362-
393-4.848MM.Kwok
4521-
5119-
5721-
63142-
6914-
7515-
815never-no SG (k square)
8721-
9343-
9914-
10522-
111-23.0MRPS
11741-
123-22.02MRPS
12933-
13519-
14111-
147441-
153-32.02MRPS
159-42.02MRPS
16522-
171-22.02MRPS
1771212-
183--2.02MRPS
189-32.02MRPS
19543-
20139-
207220-
21336-1.65MK.Wozny
219--2.102MRPS
225-never2.0Mno SG (k square)
23116-
237never17-no Twin possible
2431211-
249-82.0M
25521-
2611114-
26744-
2732-2.0M
279--2.0M
28517-
291155329-
297141-
303-62.71MNPLB
30914-
315224-
32111-
32741-
33354-2.71MNPLB
33937-
34546-
351--2.0M
35724-
36322-
369134-
37532-
381171-
38728-2.0M
393--2.0M
39911-2.0M
40511-
411113-
4172-1.59MNPLB
423822-
4291-1.59MNPLB
435413-
4411never-no SG (k square)
44724-
453482-
45938-
46565-
4713-1.59MNPLB
477-11.59MNPLB
48322-
48953-
495168-
501-21.59MNPLB
50721-
5136267-
519117-
52511-
5311-540kNPLB
53765-
543--540kNPLB
549113-
555930-
5617-540kNPLB
5672-540kNPLB
573344-540kNPLB
579-35540kNPLB
58522-
5915-540kNPLB
59770-540kNPLB
60310-540kNPLB
609193-
61511-
621322-
62765-
6333242-
6391-540kNPLB
64511-
651113-
6575829-
663243-
669118-
67552-
68131-540kNPLB
68734-540kNPLB
693--540kNPLB
6995-540kNPLB
70531-
711175-
717124-
7236-540kNPLB
729-never540kno SG (k square), NPLB
73532-
74111-
74768-
753--540kNPLB
75917-540kNPLB
76549-
771--540kNPLB
7771020-
783-3540kNPLB
789-23540kNPLB
7953-540kNPLB
801-1540kNPLB
807nevernever540kno Twin, no SG possible