The New Mersenne Prime Conjecture
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:
- p = 2k+/-1 or p = 4k+/-3
- 2p-1 is a prime (obviously a Mersenne prime)
- (2p+1)/3 is a prime.
Below we list all of the primes p where
- p < 10000000 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.)
(There is more current data at https://sites.google.com/site/bearnol/math/mersenneplustwo)
| 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: 181122707148161338644285289935461939 |
| 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: 8114899840326779533679915276470289950126585679 | 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: 513668017883326358119 | no: 13091975735977 |
| 65539 | yes (+3) | no: 3354489977369 | no: 58599599603 |
| 83339 | no | no: 166679 | yes |
| 86243 | no | yes | no: 1627710365249 |
| 95369 | no | no: 297995890279 | yes |
| 110503 | no | yes | no: 48832113344350037579071829046935480686609 |
| 117239 | no | no | yes |
| 127031 | no | no: 12194977 | yes |
| 131071 | yes (-1) | no: 231733529 | no: 2883563 |
| 132049 | no | yes | no: 618913299601153 |
| 138937 | no | no: 100068818503 | yes |
| 141079 | no | no: 458506751 | yes (prp) |
| 216091 | no | yes | no: 10704103333093885136919332089553661899 |
| 262147 | yes (+3) | no: 268179002471 | no: 4194353 |
| 267017 | no | no: 1602103 | yes (prp) |
| 269987 | no | no: 1940498230606195707774295455176153 | yes (prp) |
| 374321 | no | no | yes (prp) |
| 524287 | yes (-1) | no: 62914441 | no |
| 756839 | no | yes | no: 1640826953 |
| 859433 | no | yes | no: 1718867 |
| 986191 | no | no | yes (prp) |
| 1048573 | yes (-3) | no: 73400111 | no |
| 1257787 | no | yes | no: 20124593 |
| 1398269 | no | yes | no: 23609117451215727502931 |
| 2976221 | no | yes | no: 434313978089 |
| 3021377 | no | yes | no: 95264016811 |
| 4031399 | no | no: 8062799 | yes (prp) |
| 4194301 | yes (-3) | no: 2873888432993463577 | no: 14294177809 |
| 6972593 | no | yes | no: 142921867730820791335455211 |
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. |
This page was originally created based off Conrad Curry's excellent New Mersenne Conjecture page. Thanks to Alex Kruppa and David Broadhurst for the suggestion.
Printed from the PrimePages <t5k.org> © Reginald McLean.