20140222, 05:41  #1 
Aug 2010
Kansas
547 Posts 
Largest Mersenne Number Fully Factored?
Does anyone know what the largest fully factored Mersenne number is (only counting those with prime exponents, with factors other than one and itsself)?
So far, I've found M7853, which appears to be fully factored... 
20140222, 06:02  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{3}·3·5^{2}·11 Posts 
Not sure of the largest known but there is:
M9901: 87770464009 . 4512717821471308759.8336998551279784091551 . 1017688752041649660766793 . 25146117302614435382787771401 . 1502440689076527620360606617623599 . P2844 Last fiddled with by retina on 20140222 at 06:15 Reason: spacing 
20140222, 06:03  #3 
Romulan Interpreter
"name field"
Jun 2011
Thailand
277A_{16} Posts 
Where "does it appears"? In factorDB the tail is composite. I was running to PrimeNet to shameless pick up eventual new factors, and report them to fDB, but found out there are no new factors beside of the two trivial ones. Are you watching some cunnigham pages or so? Where?
edit: crosspost, I was talking to the OP Last fiddled with by LaurV on 20140222 at 06:19 
20140222, 06:13  #4 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
24757_{8} Posts 
Will Edgington has 9733 as being fully factored:
2932747561 * 353435802999708808999 * 4424579967215442704801447 * the cofactor is prime but not listed. And 684127 might be fully factored 23765203727 * the cofactor is at least a pseudoprime in some base other than 2 Last fiddled with by Uncwilly on 20140222 at 06:16 Reason: 684127 
20140222, 06:16  #5 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2·31·163 Posts 
Searching fdb up to the mersenne prime M23209 (wanted to go to 30k, but it became slow), it appears that M20887 is the highest FF.
edit, meantime it went through, so M26903, M28759, M28771, M29473, are all FFed (only expos below 30k tested). Last fiddled with by LaurV on 20140222 at 06:25 Reason: link 
20140222, 07:35  #6  
Aug 2010
Kansas
547 Posts 
Quote:


20140222, 08:30  #7 
Jun 2003
2^{2}·7·193 Posts 
Last fiddled with by axn on 20140222 at 08:33 
20140223, 07:17  #8 
Aug 2006
5,987 Posts 

20160522, 21:39  #9 
Sep 2003
13×199 Posts 
In factorDB, selecting "n is prime", "n is odd", "FF Show fully factored numbers", the following numbers appear:
Code:
(smaller exponents omitted) 2^77571=233293220467553594643512097574361 2^88491=52368383.15264764469472455023 2^96971=724126946527.19092282046942032847 2^97331=2932747561.353435802999708808999.4424579967215442704801447 2^99011=87770464009.4512717821471308759.8336998551279784091551.1017688752041649660766793.25146117302614435382787771401.1502440689076527620360606617623599 2^100071=240169.60282169.136255313.10368448917257 2^101691=10402314702094700470118039921523041260063 2^102111=81689.735193.5108003713569136882634199446306201 2^104331=146063.7345550506166399.17578384916225511229570561.407523153578238773059225963827711400649 2^111171=60138110048076002069201.5956230361711049200365020316257263269553 2^118131=70879.207971134271377 2^124511=4980401.15289230353.1143390212315192593598809 2^145611=8074991336582835391 2^146211=1958650799081.9787919624201558678734079 2^170291=418879343 2^176831=234000819833373807217.62265855698776681155719328257 2^191211=917809.415147656569.1531543915081.27784129616513881634842031 2^208871=694257144641.3156563122511.28533972487913.1893804442513836092687 2^269031=1113285395642134415541632833178044793 2^287591=226160777 2^287711=104726441 2^294731=5613392570256862943.24876264677503329001 2^325311=65063.25225122959 2^353391=5776625742089.291148630508887.7028028455954046211351.4153830438466899077960892137 2^412631=1402943.983437775590306674647 2^415211=2989513.249375127.55803711703045241786952239 2^416811=1052945423.16647332713153.2853686272534246492102086015457 2^571311=457049.49644668023.359585713337.7535393191738347569 2^581991=237604901713907577052391 2^637031=42808417 2^829391=867140681119.1018662740943783967 2^861371=2584111.7747937967916174363624460881 2^863711=41681512921035887 2^876911=500982892169.1610747697738457 2^1063911=286105171290931103 2^1304391=260879 2^1368831=536581361 2^1738671=52536637502689 2^2215091=292391881 2^2700591=540119.6481417.7124976157756725967 2^2712111=613961495159 2^2715491=238749682487 2^4065831=813167 2^4324571=1672739247834685086279697 2^4403991=880799.31518475633.16210820281161978209 2^4884411=61543567.30051203516986199 Of the numbers discussed in previous messages, M7853 does not appear in the list, but M9901, M9733, and the others do. In addition to the above, the Henri & Renaud link also lists the following as probable primes, which do not appear in factorDB (perhaps factorDB limits its data to n < 500000 ?): Code:
(2^34644731)/604874508299177 (2^23274171)/23915387348002001 (2^17907431)/(146840927*158358984977*3835546416767873*20752172271489035681) (2^13049831)/52199321 (2^11681831)/54763676838381762583 (2^10106231)/12602017578957977 (2^7501511)/(429934042631*7590093831289*397764574647511*8361437834787151*17383638888678527263) (2^6963431)/11141489/36009913139329 (2^6841271)/23765203727 (2^6759771)/(1686378749257*7171117283326998925471) (2^5765511)/4612409/64758208321/242584327930759 If you take all the exponents considered by factorDB to be fully factored (including the ones lower than M7757 which were omitted above for brevity), and also include the eleven additional large exponents from Henri & Renaud, then it seems there are only 301 Mersenne numbers that are either fully factored or probablyfullyfactored. Can this be correct? 
20160523, 02:18  #10 
Romulan Interpreter
"name field"
Jun 2011
Thailand
23572_{8} Posts 

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