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Old 2014-02-22, 05:41   #1
c10ck3r
 
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Default 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...
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Old 2014-02-22, 06:02   #2
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Not sure of the largest known but there is:

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

Last fiddled with by retina on 2014-02-22 at 06:15 Reason: spacing
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Old 2014-02-22, 06:03   #3
LaurV
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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 2014-02-22 at 06:19
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Old 2014-02-22, 06:13   #4
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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

Last fiddled with by Uncwilly on 2014-02-22 at 06:16 Reason: 684127
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Old 2014-02-22, 06:16   #5
LaurV
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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).

Last fiddled with by LaurV on 2014-02-22 at 06:25 Reason: link
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Old 2014-02-22, 07:35   #6
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Quote:
Originally Posted by LaurV View Post
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...
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Old 2014-02-22, 08:30   #7
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http://primes.utm.edu/top20/page.php?id=49

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

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Old 2014-02-23, 07:17   #8
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Quote:
Originally Posted by c10ck3r View Post
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.
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Old 2016-05-22, 21:39   #9
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In factorDB, selecting "n is prime", "n is odd", "FF Show fully factored numbers", the following numbers appear:

Code:
(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 ?):

Code:
(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?
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Old 2016-05-23, 02:18   #10
LaurV
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Quote:
Originally Posted by GP2 View Post
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.
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Old 2016-05-23, 02:48   #11
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See here and here for PRP-testing coordination and status
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