Re: Mersenne: M727 factored!
"Daran" <[EMAIL PROTECTED]> asks > Is M751 now the smallest unfactorised composite Mersenne? What is the > smallest Mersenne not completely factorised? M751 and M809 are the first Mersennes with no known factor. The first holes in the 2^n - 1 table are 2,673- c151 and 2,683- c203. This means, for example, that 2^673 - 1 is partially factored, but it has a 151-digit composite cofactor. The first holes in the 2^n + 1 table are 2,647+ c169 and 2,653+ c154. The first holes in the 2LM table are 2,1238L c160 and 2,1238M c145. These denote cofactors of 2^619 - 2^310 + 1 and 2^619 + 2^310 + 1, both of which divide 2^1238 + 1. Below are ten recently found factorizations. Algebraic factors, such as the factors 23 and 89 of 2^671 - 1 (these divide 2^11 - 1, which in turn divides 2^671 - 1) do not appear. Factors above the * lines were previously known. Peter Montgomery [EMAIL PROTECTED] September, 2001 C(2,619+) * c186 = p91.p96 1257388159910804265763446600825278256318012249697661907431303034629811311740864601663325811 576735513593459498091224888775105070536100466874272552892992673568274909599171018374973329229033 C(2,632+) 286297736737 * c177 = p66.p111 47121166891830156151548920824621951426679953187468148445645729 514032034228747931945571962470228019101894473686825197910636442532520433593399910044880487939432413264840902977 C(2,641+) 1283 32051 139739 353833 1078163 * c169 = p59.p110 73819843823154749726309925820314356063695778208135055585507 18795947089943685289042850201861326689719641302766796883477502913493254840480457593719603600834006204492522841 C(2,641-) 35897 4 1173835097 2401258891949526685926151441 * c148 = p69.p79 745276300734440606226386924312213175677903182797334854064486587296999 2420161564200739329410254310444778820196576654139080232429544162649795567983079 C(2,643-) 3189281 * c188 = p71.p117 22532429052605670225026391054393428833168207234802434915090881303620353 507909591297683949138862971271266635431758872031092542127980551589004038646657157217329569167343063743426799521984799 C(2,671-) 116356769 33491655209 64110547427930873 * c145 = p68.p78 13646560594525825890627182668772241639702837721889959372317451952089 608833519146176962786346063898868909094632504100539398786357475514441579020823 C(2,727-) * c219 = p98.p122 17606291711815434037934881872331611670777491166445300472749449436575622328171096762265466521858927 40099499726183758517891939428601665707063794593443940689888526556802581529262728143398959743444150539520890742947533452401 C(2,1202L) 7213 * c178 = p87.p91 322191336498946329196503049475322564558677154558189688098324958943433876457693205092709 3571063752373727959434120513220011301363863161567383982128636656862221716975343445451820353 C(2,1222L) 1363753 * c161 = p69.p92 390941529316414655423854492083019690148253103500590794058313678419233 70215922956051621713037377195110638684576079957295845924735328290270267776117780526372201309 C(2,1234M) 86381 7367588575848411802768597653205046693 * c144 = p56.p88 25030363534817185101957125006047751030454874478028239473 6828519750766734356393222104746037438762841641726469993132504189166732581463128027874013 _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
Eric Hahn wrote: > M727 - 94.3716% probability - 2 factors > M727 - 52.8693% probability - 3 factors > M727 - 6.0014% probability - 4+ factors > M727 - 91.1834% probability - 313-bit min. factor size > M727 - 93.0447% probability - 428-bit max. factor size It's impossible for these to be both true, since their product would then have 741 bits. > M727 - 21.7336% probability - highly composite factors What does this mean? > M751 - 83.8467% probability - 2 factors > M751 - 74.2974% probability - 3 factors > M751 - 19.5801% probability - 4+ factors > M751 - 87.2999% probability - 281-bit min. factor size > M751 - 81.0003% probability - 526-bit max. factor size Ditto 807 bits. > M751 - 30.1716% probability - highly composite factors Is M751 now the smallest unfactorised composite Mersenne? What is the smallest Mersenne not completely factorised? Regards Daran G. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
-Original Message- From: Alexander Kruppa <[EMAIL PROTECTED]> To: Frank Solensky <[EMAIL PROTECTED]>; [EMAIL PROTECTED] <[EMAIL PROTECTED]> Date: 04 September 2001 17:48 Subject: Re: Mersenne: M727 factored! >Frank Solensky wrote: >> >> > I have what's probably a silly question. >> > >> > You wrote this: >> > > M727 - 94.3716% probability - 2 factors >> > > M727 - 52.8693% probability - 3 factors >> > > M727 - 6.0014% probability - 4+ factors >> >> I assumed it was that they aren't mutually exclusive -- "2 factors" should >> have been "2+ factors" instead. > >Thats what I first thought, but then 2+ factors should have been 100% as >we knew that M727 is not prime. All of the probabilities are either 0% or 100%, so I would guess that these are probabilities calculated on some heuristic basis which did not take the result of the LL test into account. Presumably we can multiply each of the above by 100/94.3716 to incorporate this additional information. >Alex Daran G. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: M727 factored!
hi Frank, thanks for the message. That makes more sense. The only problem with your interpretation is that we know M727 is composite, so shouldn't the "2 factors" probability be 100%? >I assumed it was that they aren't mutually exclusive -- "2 factors" should >have been "2+ factors" instead. > Alan _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
Frank Solensky wrote: > > > I have what's probably a silly question. > > > > You wrote this: > > > M727 - 94.3716% probability - 2 factors > > > M727 - 52.8693% probability - 3 factors > > > M727 - 6.0014% probability - 4+ factors > > I assumed it was that they aren't mutually exclusive -- "2 factors" should > have been "2+ factors" instead. Thats what I first thought, but then 2+ factors should have been 100% as we knew that M727 is not prime. Alex _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
RE: Mersenne: M727 factored!
> I have what's probably a silly question. > > You wrote this: > > M727 - 94.3716% probability - 2 factors > > M727 - 52.8693% probability - 3 factors > > M727 - 6.0014% probability - 4+ factors I assumed it was that they aren't mutually exclusive -- "2 factors" should have been "2+ factors" instead. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
hi Eric (and everyone else), I have what's probably a silly question. You wrote this: > M727 - 94.3716% probability - 2 factors > M727 - 52.8693% probability - 3 factors > M727 - 6.0014% probability - 4+ factors Why do these probabilities add up to more than 100%? > M751 - 83.8467% probability - 2 factors > M751 - 74.2974% probability - 3 factors > M751 - 19.5801% probability - 4+ factors ditto thanks, Alan Simpson _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
Eric Hahn wrote: > M727 - 94.3716% probability - 2 factors > M727 - 52.8693% probability - 3 factors > M727 - 6.0014% probability - 4+ factors > M727 - 91.1834% probability - 313-bit min. factor size > M727 - 93.0447% probability - 428-bit max. factor size > M727 - 21.7336% probability - highly composite factors > M751 - 83.8467% probability - 2 factors > M751 - 74.2974% probability - 3 factors > M751 - 19.5801% probability - 4+ factors > M751 - 87.2999% probability - 281-bit min. factor size > M751 - 81.0003% probability - 526-bit max. factor size > M751 - 30.1716% probability - highly composite factors Eric, I was wondering what variables are the basis for these statistics, and whether they can be used to investigate whether a particular Mersenne number is likely to be prime? cheers, Bill. _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Re: Mersenne: M727 factored!
George Woltman wrote: >>M727, the smallest Mersenne number with no known factor, is done. >>(It was clearly out of reach of ecm.) >> >>--- Start of forwarded message --- >>From: Peter-Lawrence.Montgomery >>Date: Thu, 30 Aug 2001 03:26:19 GMT >> >>C(2,727-) >>* c219 = prp98.prp128.SNFSDodson/AKL/CWI >>* Penultimate prime champion >>* Runner-up for SNFS difficulty >> >>17606291711815434037934881872331611670777491166445300472749449436575622328 171096762265466521858927 >> >>40099499726183758517891939428601665707063794593443940689888526556802581529 262728143398959743444150539520890742947533452401 >>--- End of forwarded message --- This result does not surprise ME in the least... Anybody on the list that saw my post back in July of last year... knows that from the statistcal anaylsis that was done at that time... I posted the following for M727: M727 - 94.3716% probability - 2 factors M727 - 52.8693% probability - 3 factors M727 - 6.0014% probability - 4+ factors M727 - 91.1834% probability - 313-bit min. factor size M727 - 93.0447% probability - 428-bit max. factor size M727 - 21.7336% probability - highly composite factors Now we know that M727 has 2 factors... and the factors are 326 and 426 bits in length, respectively... Preliminary testing also shows that ( factor - 1 ) is NOT highly composite (having many, many factors)... Would anybody care to verify the data I posted back then for M751 M751 - 83.8467% probability - 2 factors M751 - 74.2974% probability - 3 factors M751 - 19.5801% probability - 4+ factors M751 - 87.2999% probability - 281-bit min. factor size M751 - 81.0003% probability - 526-bit max. factor size M751 - 30.1716% probability - highly composite factors Eric _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
Mersenne: M727 factored!
>M727, the smallest Mersenne number with no known factor, is done. >(It was clearly out of reach of ecm.) > >--- Start of forwarded message --- >From: Peter-Lawrence.Montgomery >Date: Thu, 30 Aug 2001 03:26:19 GMT > >C(2,727-) >* c219 = prp98.prp128.SNFSDodson/AKL/CWI >* Penultimate prime champion >* Runner-up for SNFS difficulty > >17606291711815434037934881872331611670777491166445300472749449436575622328171096762265466521858927 > >40099499726183758517891939428601665707063794593443940689888526556802581529262728143398959743444150539520890742947533452401 >--- End of forwarded message --- _ Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
