Largest Mersenne Number Fully Factored?

[c10ck3r]

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...

[retina]

Not sure of the largest known but there is:

M9901:
87770464009 . 4512717821471308759.8336998551279784091551 . 1017688752041649660766793 . 25146117302614435382787771401 . 1502440689076527620360606617623599 . P2844

[LaurV]

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

[Uncwilly]

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 pseudo-prime in some base other than 2

[LaurV]

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 FF-ed (only expos below 30k tested).

[c10ck3r]

Originally Posted by LaurV

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

Used alpertron's app, it called the last deal prime ig...

[axn]

http://primes.utm.edu/top20/page.php?id=49

http://www.primenumbers.net/prptop/s...&action=Search

[CRGreathouse]

Originally Posted by c10ck3r

Does anyone know what the largest fully factored Mersenne number is (only counting those with prime exponents, with factors other than one and itsself)?

To make axn's answer more explicit: M63703 factors as 42808417 times a 19,169-digit prime.

[GP2]

In factorDB, selecting "n is prime", "n is odd", "FF Show fully factored numbers", the following numbers appear:

(smaller exponents omitted)

2^7757-1=233293220467553594643512097574361
2^8849-1=52368383.15264764469472455023
2^9697-1=724126946527.19092282046942032847
2^9733-1=2932747561.353435802999708808999.4424579967215442704801447
2^9901-1=87770464009.4512717821471308759.8336998551279784091551.1017688752041649660766793.25146117302614435382787771401.1502440689076527620360606617623599
2^10007-1=240169.60282169.136255313.10368448917257
2^10169-1=10402314702094700470118039921523041260063
2^10211-1=81689.735193.5108003713569136882634199446306201
2^10433-1=146063.7345550506166399.17578384916225511229570561.407523153578238773059225963827711400649
2^11117-1=60138110048076002069201.5956230361711049200365020316257263269553
2^11813-1=70879.207971134271377
2^12451-1=4980401.15289230353.1143390212315192593598809
2^14561-1=8074991336582835391
2^14621-1=1958650799081.9787919624201558678734079
2^17029-1=418879343
2^17683-1=234000819833373807217.62265855698776681155719328257
2^19121-1=917809.415147656569.1531543915081.27784129616513881634842031
2^20887-1=694257144641.3156563122511.28533972487913.1893804442513836092687
2^26903-1=1113285395642134415541632833178044793
2^28759-1=226160777
2^28771-1=104726441
2^29473-1=5613392570256862943.24876264677503329001
2^32531-1=65063.25225122959
2^35339-1=5776625742089.291148630508887.7028028455954046211351.4153830438466899077960892137
2^41263-1=1402943.983437775590306674647
2^41521-1=2989513.249375127.55803711703045241786952239
2^41681-1=1052945423.16647332713153.2853686272534246492102086015457
2^57131-1=457049.49644668023.359585713337.7535393191738347569
2^58199-1=237604901713907577052391
2^63703-1=42808417

2^82939-1=867140681119.1018662740943783967
2^86137-1=2584111.7747937967916174363624460881
2^86371-1=41681512921035887
2^87691-1=500982892169.1610747697738457
2^106391-1=286105171290931103
2^130439-1=260879
2^136883-1=536581361
2^173867-1=52536637502689
2^221509-1=292391881
2^270059-1=540119.6481417.7124976157756725967
2^271211-1=613961495159
2^271549-1=238749682487
2^406583-1=813167
2^432457-1=1672739247834685086279697
2^440399-1=880799.31518475633.16210820281161978209
2^488441-1=61543567.30051203516986199

I checked systematically up to n=64000, and then looked up the rest based on the Henri & Renaud link given in a previous message.

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 ?):

(2^3464473-1)/604874508299177
(2^2327417-1)/23915387348002001
(2^1790743-1)/(146840927*158358984977*3835546416767873*20752172271489035681)
(2^1304983-1)/52199321
(2^1168183-1)/54763676838381762583
(2^1010623-1)/12602017578957977
(2^750151-1)/(429934042631*7590093831289*397764574647511*8361437834787151*17383638888678527263)
(2^696343-1)/11141489/36009913139329
(2^684127-1)/23765203727
(2^675977-1)/(1686378749257*7171117283326998925471)
(2^576551-1)/4612409/64758208321/242584327930759

However the Primo top 20 page and the Prime Pages suggest that only the ones up to n=63703 have actually been proven prime via ECPP and the cofactors for exponent values higher than that are only probable primes.

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 probably-fully-factored. Can this be correct?

[LaurV]

Originally Posted by GP2

then it seems there are only 301 Mersenne numbers that are either fully factored or probably-fully-factored. Can this be correct?

Correct. The number may or may not be right, I didn't count them, but you got the procedure right.

[axn]

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