The New Mersenne Prime Conjecture 
(Another of the Prime Pages' resources)
 New record prime: 274,207,281-1 with 22,338,618 digits by Cooper, Woltman, Kurowski, Blosser & GIMPS (7 Jan 2016).


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Leonhard Euler showed:
Theorem:  If  k>1 and  p=4k+3  is prime, then 2p+1 is prime if and only if 2p+1 divides 2p-1.
It is also clear that if p is an odd composite, then 2p-1 and (2p+1)/3 are composite.  Looking at these theorems and various numerical results, Bateman, Selfridge and Wagstaff (The new Mersenne conjecture, Amer. Math. Monthly, 96 (1989) 125-128 [BSW89]) have conjectured the following.

Conjecture (*): Let p be any odd natural number.  If two of the following conditions hold, then so does the third:
  1. p = 2k+/-1   or   p = 4k+/-3
  2. 2p-1 is a prime (obviously a Mersenne prime)
  3. (2p+1)/3 is a prime.

Below we list all of the primes p where

  •   p < 20000000 and p = 2k ± 1 or p = 4k ± 3.
  •  2p - 1 is known to be prime.
  •  (2p + 1)/3 is known to be prime or a probable-prime.

Details are in the table and notes below.  (The conjecture is that if any two of the entries in a row is yes, so is the third.)

p p = 2k±1 or 4k±3? 2p - 1 prime? (2p + 1)/3 prime?
3 yes (-1) yes yes
5 yes (+1) yes yes
7 yes (-1 or +3) yes yes
11 no no: 23 yes
13 yes (-3) yes yes
17 yes (+1) yes yes
19 yes (+3) yes yes
23 no no: 47 yes
31 yes (-1) yes yes
43 no no: 431 yes
61 yes (-3) yes yes
67 yes (+3) no: 193707721 no: 7327657
79 no no: 2687 yes
89 no yes no: 179
101 no no: 7432339208719 yes
107 no yes no: 643
127 yes (-1) yes yes
167 no no: 2349023 yes
191 no no: 383 yes
199 no no: 164504919713 yes
257 yes (+1) no: 535006138814359 no: 37239639534523
313 no no: 10960009 yes
347 no no: 14143189112952632419639 yes
521 no yes no: 501203
607 no yes no: 115331
701 no no: 796337 yes
1021 yes (-3) no: 40841 no: 10211
1279 no yes no: 706009
1709 no no: 379399 yes [Morain1990a]
2203 no yes no: 13219
2281 no yes no: 22811
2617 no no: 78511 yes [Morain1990a]
3217 no yes no: 7489177
3539 no no: 7079 yes [Morain1990a]
4093 yes (-3) no: 2397911088359 no: 3732912210059
4099 yes(+3) no: 73783 no: 2164273
4253 no yes no: 118071787
4423 no yes no: 2827782322058633
5807 no no: 139369 yes (note 6)
8191 yes (-1) no: 338193759479 no
9689 no yes no: 19379
9941 no yes no: 11120148512909357034073
10501 no no: 2160708549249199 yes (note 5)
10691 no no: 21383 yes (note 1)
11213 no yes no
11279 no no: 2198029886879 yes (note 4)
12391 no no: 198257 yes (note 3)
14479 no no: 27885728233673 yes (note 2)
16381 yes (-3) no no: 163811
19937 no yes no
21701 no yes no: 43403
23209 no yes no: 4688219
42737 no no yes (note 7)
44497 no yes no: 2135857
65537 yes (+1) no no: 13091975735977
65539 yes (+3) no: 3354489977369 no: 58599599603
83339 no no: 166679 yes (prp)
86243 no yes no: 1627710365249
95369 no no: 297995890279 yes (prp)
110503 no yes no
117239 no no yes (prp)
127031 no no: 12194977 yes (prp)
131071 yes (-1) no: 231733529 no: 2883563
132049 no yes no: 618913299601153
138937 no no: 100068818503 yes (prp)
141079 no no: 458506751 yes (prp)
216091 no yes no
262147 yes (+3) no: 268179002471 no: 4194353
267017 no no: 1602103 yes (prp)
269987 no no yes (prp)
374321 no no yes (prp)
524287 yes (-1) no: 62914441 no
756839 no yes no: 1640826953
859433 no yes no: 1718867
1048573 yes (-3) no: 73400111 no
1257787 no yes no: 20124593
1398269 no yes no
2976221 no yes no: 434313978089
3021377 no yes no: 95264016811
4194301 yes (-3) no: 2873888432993463577 no: 14294177809
6972593 no yes no
13466917 no yes no: 781081187
16777213 yes (-3) no no: 68470872139190782171 (note 8)
20996011 no yes unknown (no factor <10^18)
24036583 no yes no: 11681779339
25964951 no yes no: 155789707
30402457 no yes unknown (no factor < 10^18)
When a small factor is known we listed it above.

Notes:

## prime note
* (any) The integers listed after 'no' are small factors of the corresponding composite.
** (any) The expression prp means probable-prime
1 10691 ECPP primality proof by David Broadhurst via Primo, certificate n10691.zip
2 14479 ECPP primality proof by David Broadhurst via Primo, certificate n14479.zip
3 12391 Proof by François Morain, see his notes.
4 11279 Proof by Preda Mihailescu, see his notes.
5 10501 Proof by François Morain, see his notes.
6 5807 Proof by Preda Mihailescu, see his notes.
7 42737 Proof by François Morain, see his notes on the prime's page.
8 16777213 Factor found by Andreas Höglund (July 2009).
     

This page is updated from Conrad Curry's excellent New Mersenne Conjecture page, originally hosted at http://orca.st.usm.edu/~cwcurry/NMC.html, but now missing.  Thanks to Alex Kruppa and David Broadhurst for suggestion.
The Prime Pages
Another prime page by Chris K. Caldwell <caldwell@utm.edu>