Mersenne cofactor

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The Prime Pages

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. Comments and suggestions requested.

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 (2⁴¹⁵²¹ - 1)/41602235382028197528613357724450752065089 12459 c54 Aug 2012 Mersenne cofactor, ECPP
2 (2⁴¹²⁶³ - 1)/(1402943 · 983437775590306674647) 12395 c59 Oct 2012 Mersenne cofactor, ECPP
3 (2³²⁵³¹ - 1)/(65063 · 25225122959) 9778 c60 Sep 2012 Mersenne cofactor, ECPP
4 (2²⁹⁴⁷³ - 1)/(5613392570256862943 · 24876264677503329001) 8835 c59 Sep 2012 Mersenne cofactor, ECPP
5 (2²⁸⁷⁷¹ - 1)/104726441 8653 c56 Feb 2012 Mersenne cofactor, ECPP
6 (2²⁸⁷⁵⁹ - 1)/226160777 8649 c60 Sep 2012 Mersenne cofactor, ECPP
7 (2²⁶⁹⁰³ - 1)/1113285395642134415541632833178044793 8063 c55 Dec 2011 Mersenne cofactor, ECPP
8 (2²⁰⁸⁸⁷ - 1)/(694257144641 · 3156563122511 · 28533972487913 · 1893804442513836092687) 6229 c4 Sep 2009 Mersenne cofactor, ECPP
9 (2¹⁷⁶⁸³ - 1)/(234000819833373807217 · 62265855698776681155719328257) 5274 c4 Sep 2009 Mersenne cofactor, ECPP
10 (2¹⁷⁰²⁹ - 1)/418879343 5118 c8 Sep 2006 Mersenne cofactor, ECPP
11 (2¹⁴⁵⁶¹ - 1)/8074991336582835391 4365 c8 Oct 2004 Mersenne cofactor, ECPP
12 (2¹⁴⁶²¹ - 1)/(1958650799081 · 9787919624201558678734079) 4365 c4 Nov 2008 Mersenne cofactor, ECPP
13 (2¹²⁴⁵¹ - 1)/(4980401 · 15289230353 · 1143390212315192593598809) 3708 c4 Oct 2008 Mersenne cofactor, ECPP
14 (2¹¹⁸¹³ - 1)/(70879 · 207971134271377) 3537 c8 Aug 2002 Mersenne cofactor, ECPP
15 (2¹¹¹¹⁷ - 1)/3581964369642\
70608221221853970927519972222557196875442622337153 3284 c4 Aug 2011 Mersenne cofactor, ECPP
16 (2¹⁰²¹¹ - 1)/306772303457009724362047724636324707614338377 3030 c4 Apr 2010 Mersenne cofactor, ECPP
17 (2¹⁰¹⁶⁹ - 1)/10402314702094700470118039921523041260063 3022 c8 Jul 2002 Mersenne cofactor, ECPP
18 (2¹⁰⁰⁰⁷ - 1)/(14477908246561 · 136255313 · 10368448917257) 2979 c8 Jun 2002 Mersenne cofactor, ECPP
19 (2⁹⁶⁹⁷ - 1)/(724126946527 · 19092282046942032847) 2888 c8 Jun 2002 Mersenne cofactor, ECPP
20 (2⁹⁷³³ - 1)/(2932747561 · 353435802999708808999 · 4424579967215442704801447) 2876 c4 Aug 2001 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