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 (2²⁰⁸⁸⁷-1)/(694257144641 · 3156563122511 · 28533972487913 · 1893804442513836092687) 6229 c4 Sep 2009 Mersenne cofactor, ECPP
2 (2¹⁷⁶⁸³-1)/(234000819833373807217 · 62265855698776681155719328257) 5274 c4 Sep 2009 Mersenne cofactor, ECPP
3 (2¹⁷⁰²⁹-1)/418879343 5118 c8 Sep 2006 Mersenne cofactor, ECPP
4 (2¹⁴⁵⁶¹-1)/8074991336582835391 4365 c8 Oct 2004 Mersenne cofactor, ECPP
5 (2¹⁴⁶²¹-1)/(1958650799081 · 9787919624201558678734079) 4365 c4 Nov 2008 Mersenne cofactor, ECPP
6 (2¹²⁴⁵¹-1)/(4980401 · 15289230353 · 1143390212315192593598809) 3708 c4 Oct 2008 Mersenne cofactor, ECPP
7 (2¹¹⁸¹³-1)/(70879 · 207971134271377) 3537 c8 Aug 2002 Mersenne cofactor, ECPP
8 (2¹⁰¹⁶⁹-1)/10402314702094700470118039921523041260063 3022 c8 Jul 2002 Mersenne cofactor, ECPP
9 (2¹⁰⁰⁰⁷-1)/(14477908246561 · 136255313 · 10368448917257) 2979 c8 Jun 2002 Mersenne cofactor, ECPP
10 (2⁹⁶⁹⁷-1)/(724126946527 · 19092282046942032847) 2888 c8 Jun 2002 Mersenne cofactor, ECPP
11 (2⁹⁷³³-1)/(2932747561 · 353435802999708808999 · 4424579967215442704801447) 2876 c4 Aug 2001 Mersenne cofactor, ECPP
12 (2⁹⁹⁰¹-1)/(87770464009 · 4512717821471308759 · 8336998551279784091551 · 1017688752041649660766793 · 25146117302614435382787771401 · 1502440689076527620360606617623599) 2844 c2 Dec 2001 Mersenne cofactor, ECPP
13 (2⁸⁸⁴⁹-1)/(52368383 · 15264764469472455023) 2637 c4 Aug 2001 Mersenne cofactor, ECPP
14 (2⁷⁷⁵⁷-1)/233293220467553594643512097574361 2303 c2 Mar 2001 Mersenne cofactor, ECPP
15 (2⁷⁶⁷³-1)/2563193011919 2298 M1 Dec 1997 Mersenne cofactor, cyclotomy
16 (2⁷³³¹-1)/458072843161 2196 EM Oct 1997 Mersenne cofactor, ECPP
17 (2⁷⁴¹⁷-1)/(1930694161304071 · 3888241452787718190543521) 2193 c8 Sep 2002 Mersenne cofactor, ECPP
18 (2⁷⁰³⁹-1)/(1324401538053479 · 8541573097 · 218216841131937276721) 2074 c2 Jan 2001 Mersenne cofactor, ECPP
19 (2⁶⁸⁸³-1)/(1885943 · 2043031664890199) 2051 M1 Dec 1997 Mersenne cofactor, cyclotomy
20 (2⁶¹⁹⁹-1)/(46113927071 · 3895469424045161025776010136475884556282201) 1813 c8 Feb 2005 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