Mersenne cofactor

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.

This page is about one of those forms.

Definitions and Notes

Mersenne primes have always been the most sought after primes! Euclid famously connected them to perfect numbers in his Elements over two millennia ago. Today tens of thosands of individuals search for the next Mersenne prime as part of GIMPS.

When searching for new Mersennes, you should always first look for small divisors (called Mersenne divisors), before applying the definitive (but time consuming) Lucas-Lehmer test. These divisors must have a very special form as Fermat and Euler proved:

Theorem:
Let p and q be odd primes. If p divides M_q, then p = 1 (mod q) and p = +/-1 (mod 8).

Sometimes, just the fact that a number divides a Mersenne is enough to show it is prime:

Theorem.
Let p = 3 (mod 4) be prime. 2p+1 is also prime if and only if 2p+1 divides M_p.

Well call a prime a Mersenne cofactor if it is a proper divisor of a Mersenne number M_p with p prime. Below we list the twenty largest known Mersenne cofactors.

Record Primes of this Type

rank prime digits who when comment

1 (2¹⁰⁶³⁹¹ - 1)/286105171290931103 32010 c95 Apr 2022 Mersenne cofactor, ECPP
2 (2⁸⁷⁶⁹¹ - 1)/806957040167570408395443233 26371 E1 Oct 2022 Mersenne cofactor, ECPP
3 (2⁸⁶³⁷¹ - 1)/41681512921035887 25984 E2 Jun 2022 Mersenne cofactor, ECPP
4 (2⁸⁶¹³⁷ - 1)/2584111 / 7747937967916174363624460881 25896 c84 May 2022 Mersenne cofactor, ECPP
5 (2⁸⁴²¹¹ - 1)/1347377 / 31358793176711980763958121 / 3314641676042347824169591561 25291 c95 Sep 2020 Mersenne cofactor, ECPP
6 (2⁸²⁹³⁹ - 1)/883323903012540278033571819073 24938 c84 Feb 2021 Mersenne cofactor, ECPP
7 (2⁷⁸⁷³⁷ - 1)/1590296767505866614563328548192658003295567890593 23654 E2 May 2022 Mersenne cofactor, ECPP
8 (2⁶³⁷⁰³ - 1)/42808417 19169 c59 Jan 2014 Mersenne cofactor, ECPP
9 (2⁵⁸¹⁹⁹ - 1)/237604901713907577052391 17497 c59 Jan 2015 Mersenne cofactor, ECPP
10 (2⁵⁷¹³¹ - 1)/61481396117165983261035042726614288722959856631 17152 c59 Dec 2015 Mersenne cofactor, ECPP
11 (2⁵³³⁸¹ - 1)/15588960193 / 38922536168186976769 / 155991271597169062945033668006103 16008 c84 Mar 2017 Mersenne cofactor, ECPP
12 (2⁵¹⁴⁸⁷ - 1)/57410994232247 / 17292148963401772464767849635553 15455 c77 Nov 2018 Mersenne cofactor, ECPP
13 (2⁴¹⁶⁸¹ - 1)/1052945423 / 16647332713153 / 2853686272534246492102086015457 12495 c77 Apr 2015 Mersenne cofactor, ECPP
14 (2⁴¹⁵²¹ - 1)/41602235382028197528613357724450752065089 12459 c54 Aug 2012 Mersenne cofactor, ECPP
15 (2⁴¹²⁶³ - 1)/(1402943 · 983437775590306674647) 12395 c59 Oct 2012 Mersenne cofactor, ECPP
16 (2³⁵³³⁹ - 1)/490988430384989040283954404\
86235033667674267835480981233904512709297747031041 10562 c77 Apr 2015 Mersenne cofactor, ECPP
17 (2³²⁵³¹ - 1)/(65063 · 25225122959) 9778 c60 Sep 2012 Mersenne cofactor, ECPP
18 (2³²⁶¹¹ - 1)/15148007312464299210917787487318\
99943932296901864652928732838910515860494755367311 9736 c90 Jul 2018 Mersenne cofactor, ECPP
19 (2²⁹⁴⁷³ - 1)/(5613392570256862943 · 24876264677503329001) 8835 c59 Sep 2012 Mersenne cofactor, ECPP
20 (2²⁸⁷⁷¹ - 1)/104726441 8653 c56 Feb 2012 Mersenne cofactor, ECPP

References

Ehrman1967
J. Ehrman, "The number of prime divisors of certain Mersenne numbers," Math. Comp., 21 (1967) 700--704. MR0223320
EKP1991
P. Erdös, P. Kiss and C. Pomerance, "On prime divisors of Mersenne numbers," Acta. Arith., 57 (1991) 267--281. MR 92d:11104
Karst1961
E. Karst, "New factors of Mersenne numbers," Math. Comp., 15 (1961) 51. MR0116481
Karst1962
E. Karst, "Search limits on divisors of Mersenne Numbers," Nordisk Tidskr. Informations-Behandling, 2 (1962) 224--227. MR0166144
Kravitz1961
S. Kravitz, "Divisors of Mersenne numbers 10,000<p<15,000," Math. Comp., 15 (1961) 292--293. MR0123508
Pomerance1986
C. Pomerance, "On primitive divisors of Mersenne numbers," Acta Arith., 46:4 (1986) 355--367. MR871278
SK1967
D. Shanks and S. Kravitz, "On the distribution of Mersenne divisors," Math. Comp., 21 (1967) 97--101. MR0220665
Wagstaff1988
S. S. Wagstaff, Jr., "Divisors of Mersenne numbers," Math. Comp., 40:161 (1983) 385--397. MR679454