Status of M(p) - 2 for all M(p) that are Mersenne primes. $Id: M3status.txt,v 1.39 2006/04/07 22:19:16 wedgingt Exp $ This is a list of the exponent and factors or primality of M(p) - 2. The intent is to see if there are any twin primes involving a Mersenne prime, M(p) = 2^p - 1. For p = 4253, 11213, and 13466917, there are no known factors but SPRP tests show M(p) - 2 is composite. Almost all of the first factors are from Conrad Curry's message to the Mersenne mailing list on 27 March 1998; I have added second and later factors for p < 32. --Will Edgington, wedgingt@acm.org, 28 March 1998 Warut Roonguthai sent me a large number of second & higher factors and the digit lengths of the corresponding composite cofactors on April 3rd. Jim Howell sent a lot of new info 1 Dec 1999, as noted. Alexander Kruppa sent some new factors on 27 Sep 2002, adding that all of them have now been tested for factors thru 10^9. Dario Alpern has a program of his own and is presently (April 2006) running it for many of these, including the one related to the largest Mersenne prime. On 24 Feb 2004, he pointed out that: Since 6 is a quadratic residue mod all primes of 2^p-3, the values of p are restricted to 1, 5, 19 and 23 mod 24 (compare this to 1, 7, 17 and 23 mod 24 for 2^p-1). On 2005 March 19, he wrote that he's completed the search thru 10^13. p primality or known factors - -------------------------- 3 prime (5) 5 prime (29) 7 5^3 13 19 431 17 53 2473 19 5 23 47 97 31 5 19 22605091 61 29 79511827903920481 89 29 191 111747611417708997553631 107 5^4 23 4253 64171 199454329 207359542541 127 5^2 213156431 31927947500766558008599290859 521 5296777 6339704466311 1265870632744692366709980462123357518003195778883 # Alexander Kruppa 8 Oct 2002 SNFS p90 # Alexander Kruppa 8 Oct 2002 607 5^5 19 263 1401457092703061 254605005861515627419 1107375018476949027223897414997873720048902633664994028034804453 p77 # Luigi Morelli and Alexander Kruppa SNFS 16 Dec 2004 per Dario Alpern 1279 5 1250902825725775007 431867767289311159291 C346 ECM 1100 curves @ bound 1e6 # Dario Alpern 3 Aug 2004 GMP-ECM 5.0.3 ECM 5200 curves @ bound 3e6 # Reto Keiser 1 Mar 2005 Prime95 per Dario Alpern ECM 8825 curves @ bound 11e6 # Dario Alpern 13 Jul 2005 ECM 1800 curves @ bound 11e6 # Geoffrey Reynolds 13 Jul 2005 2203 5 C663 ECM 1100 curves @ bound 1e6 # Dario Alpern 3 Aug 2004 GMP-ECM 5.0.3 ECM 600 curves @ bound 3e6 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 4500 curves @ bound 3e6 # Dario Alpern 1 Mar 2005 Prime95 ECM 200 curves @ bound 11e6 # "wolf" 1 Mar 2005 Prime95 per Dario Alpern ECM 1717 curves @ bound 11e6 # Geoffrey Reynolds 24 Oct 2005 per Dario Alpern ECM 9000 curves @ bound 11e6 # Dario Alpern 24 Oct 2005 2281 19 110063 130349 300667 16806649 42437987466701 33449319205008787 C633 ECM 1100 curves @ bound 1e6 # Dario Alpern 3 Aug 2004 GMP-ECM 5.0.3 ECM 400 curves @ bound 3e6 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 4845 curves @ bound 3e6 # Dario Alpern 1 Mar 2005 Prime95 ECM 1850 curves @ bound 11e6 # Geoffrey Reynolds 24 Oct 2005 Prime95 3217 19^2 152429799520123 C952 ECM 1100 curves @ bound 1e6 # Dario Alpern 3 Aug 2004 GMP-ECM 5.0.3 ECM 400 curves @ bound 3e6 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 4668 cruves @ bound 3e6 # Dario Alpern 2 May 2005 ECM 1562 curves @ bound 11e6 # Geoffrey Reynolds 24 Oct 2005 Prime95 4253 C1281 base-2 Fermat test residue is 0x049F1774733C2358 Warut Roonguthai ECM 300 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 1100 curves @ bound 1e6 # Dario Alpern 12 Oct 2004 GMP-ECM 5.0.3 ECM 5100 curves @ bound 3e6 # Dario Alpern 1 Mar 2005 Prime95 ECM 500 curves @ bound 11e6 # "wolf" 2 May 2005 Prime95 per Dario Alpern ECM 2579 curves @ bound 11e6 # Dario Alpern 24 Oct 2005 Prime95 4423 5 19 6793 4045590660829172226299709913 # Dario Alpern 25 Aug 2004 GMP-ECM C4 C1299 # Dario Alpern 2 Sep 2004 ECM 300 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 1100 curves @ bound 1e6 # Dario Alpern 20 Oct 2004 GMP-ECM 5.0.3 ECM 400 curves @ bound 3e6 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 4670 curves @ bound 3e6 # Dario Alpern 2 May 2005 Prime95 ECM 154 curves @ bound 11e6 # Dario Alpern 24 Oct 2005 Prime95 9689 53 10513 30403 # Jim Howell 1 Dec 1999 413713 424079 # Alexander Kruppa 27 Sep 2002 13756502687 # Dario Alpern 24 Nov 2003 184105442683 # Dario Alpern 27 Nov 2003 82925904625558829 # Dario Alpern 15 Dec 2003 492257955547709423 # Dario Alpern 16 Dec 2003 591191512024933363079 # Dario Alpern 16 Dec 2003 ECM 300 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 254 curves @ bound 1e6 # Dario Alpern 12 Oct 2004 GMP-ECM 5.0.3 ECM 500 curves @ bound 3e6 # Geoffrey Reynolds 13 Jul 2005 Prime95 per Dario Alpern ECM 4696 curves @ bound 3e6 # Dario Alpern 2 May 2005 Prime95 9941 23 46679621 c2984 # Alexander Kruppa 27 Sep 2002 813319063520207 # Dario Alpern 15 Dec 2003 95215207883454642475856743478881558723 # Dario Alpern 2 May 2005 ECM 300 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 245 curves @ bound 1e6 # Dario Alpern 25 Aug 2004 GMP-ECM 5.0.3 ECM 500 curves @ bound 3e6 # Geoffrey Reynolds 13 Jul 2005 Prime95 per Dario Alpern ECM 4688 curves @ bound 3e6 # Dario Alpern 2 May 2005 Prime95 11213 C3376 base-2 Fermat residue 0x42D4FFEEDD386C23 # Jim Howell 1 Dec 1999 ECM 203 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 502 curves @ bound 25e4 # Dario Alpern 12 Oct 2004 GMP-ECM 5.0.3 ECM 1900 curves @ bound 1e6 # Dario Alpern 1 Mar 2005 Prime95 ECM 100 curves @ bound 3e6 @ "wolf" 2 May 2005 Prime95 per Dario Alpern ECM 4984 curves @ bound 3e6 # Dario Alpern 13 Jul 2005 Prime95 19937 47 225509 # Alexander Kruppa 27 Sep 2002 79030170210803391941 # Dario Alpern 08 Jan 2004 191224234861533507787 # Dario Alpern 08 Jan 2004 ECM 250 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 509 curves @ bound 25e4 # Dario Alpern 13 Dec 2004 GMP-ECM 5.0.3 ECM 1900 curves @ bound 1e6 # Dario Alpern 13 Jul 2005 Prime95 21701 53 7340492399 # Dario Alpern 24 Nov 2003 1661273513193434263801595913416669 # Dario Alpern 7 June 2005 Prime95 v24.6 ECM ECM 67 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 492 curves @ bound 25e4 # Dario Alpern 13 Dec 2004 GMP-ECM 5.0.3 ECM 1900 curves @ bound 1e6 # Dario Alpern 13 Jul 2005 Prime95 23209 53 ECM 305 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 700 curves @ bound 25e4 # Dario Alpern 13 Dec 2004 Prime95 version 24.6 ECM 1900 curves @ bound 1e6 # Dario Alpern 13 Jul 2005 Prime95 44497 29 13249 # Jim Howell 1 Dec 1999 279324722047118006717 # Dario Alpern 15 Jan 2004 ECM 247 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 700 curves @ bound 25e4 # Dario Alpern 15 Dec 2004 Prime95 version 24.6 ECM 1900 curves @ bound 1e6 # Dario Alpern 24 Oct 2005 Prime95 86243 5 406176499 # Alexander Kruppa 27 Sep 2002 3486862033 # Alexander Kruppa 22 Jan 2003 C25943 # Alexander Kruppa 22 Jan 2003 ECM 69 curves @ bound 5e4 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 700 curves @ bound 25e4 # Dario Alpern 1 Mar 2005 Prime95 version 24.6 110503 5 23 211 # Jim Howell 1 Dec 1999 267285647918587 # Dario Alpern 19 Jan 2004 17618226202282711821506083 # Dario Alpern 13 Dec 2004 11343032172898797250418606131 # Dario Alpern 13 Dec 2004 ECM 700 curves @ bound 5e4 # Dario Alpern 13 Jul 2005 Prime95 version 24.6 132049 17040433 ECM 143 curves @ bound 11e3 # Dario Alpern 13 Dec 2004 Prime95 version 24.6 ECM 300 curves @ bound 5e4 # Dario Alpern 15 Dec 2004 Prime95 version 24.6 ECM 700 curves @ bound 25e4 # Dario Alpern 13 Jul 2005 Prime95 version 24.6 216091 5 58611307 # Alexander Kruppa 27 Sep 2002 111567943033 # Dario Alpern 24 Nov 2003 26145857636113872067 # Dario Alpern 15 Dec 2004 ECM 25 curves @ bound 11e3 # Dario Alpern 3 Aug 2004 GMP-ECM 4C ECM 300 curves @ bound 5e4 # Dario Alpern 1 Mar 2005 Prime95 version 24.6 ECM 700 curves @ bound 25e4 # Dario Alpern 24 Oct 2005 Prime95 version 24.6 756839 5 17597 # Jim Howell 1 Dec 1999 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program 55648325182377547 # Geoffrey Reynolds 9 Dec 2004 Prime95 verson 24.6 ECM 300 curves @ bound 5e4 # Dario Alpern 2 May 2005 Prime95 version 24.6 859433 4219 7672260481 # Dario Alpern 24 Nov 2003 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 90 curves @ bound 11e3 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 286 curves @ bound 5e4 # Dario Alpern 2 May 2005 1257787 5 5 # Dario Alpern 08 Aug 2005 45677 # Alexander Kruppa 27 Sep 2002 3845874014189 # Dario Alpern 24 Feb 2004 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program 189128612441688911 # Geoffrey Reynolds 9 Dec 2004 Prime95 verson 24.6 ECM 90 curves @ bound 11e3 # Geoffrey Reynolds 1 Mar 2005 Prime95 per Dario Alpern ECM 300 curves @ bound 5e4 # Dario Alpern 24 Oct 2005 Prime95 1398269 29 101 # Alexander Kruppa 27 Sep 2002 52768191431 # Dario Alpern 24 Nov 2003 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 90 curves @ bound 11e3 # Dario Alpern 1 Mar 2005 Prime95 2976221 293 7418423 # Alexander Kruppa 27 Sep 2002 973870258997 # Dario Alpern 11 Dec 2003 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 90 curves @ bound 11e3 # Dario Alpern 1 Mar 2005 Prime95 3021377 10513 570195289943 # Dario Alpern 02 Dec 2003 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 90 curves @ bound 11e3 # Dario Alpern 1 Mar 2005 Prime95 6972593 29 53 4297 # Jim Howell, 1 Dec 1999 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program 479604459848711 # Dario Alpern 17 Dec 2004 Prime95 v24.6 B1=2000 B2=200000 ECM 90 curves @ bound 11e3 # Dario Alpern 1 Mar 2005 Prime95 13466917 11336430683041328998589 # Alexander Kruppa 24 Oct 2005 Prime95 ECM stage 1 50000 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 90 curves @ bound 11e3 # Alexander Kruppa 13 Jul 2005 Prime95 per Dario Alpern ECM 12 curves @ bound 5e4 # Alexander Kruppa 24 Oct 2005 Prime95 per Dario Alpern 20996011 5 # Dario Alpern 02 Dec 2003 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 20 curves @ bound 2e3 # Dario Alpern 1 Mar 2005 Prime95 24036583 5 19 47 431 # Alexander Kruppa 31 May 2004 19 # (repeated factor) Dario Alpern 13 July 2005 trial division thru 21151190000000 # Dario Alpern 25 Aug 2004 own program ECM 20 curves @ bound 2e3 # Dario Alpern 13 Jul 2005 Prime95 active ECM 62 curves @ bound 11e3 # Dario Alpern 24 Oct 2005 Prime95 active 25964951 5 2742407 # Dario Alpern 1 Mar 2005 Prime95 ECM 20 curves @ bound 2e3 # Dario Alpern 2 May 2005 Prime95