The 343rd fully-factored or probably-fully-factored Mersenne number with prime exponent (not including the Mersenne primes themselves) is [M]M2749[/M].

The most recent factor (65 digits!) was found by Ryan Propper on January 28 and the PRP test was done by Jinyuan Wang.

The cofactor has already been certified prime.

[URL="http://factordb.com/index.php?id=1100000001440761806"]FactorDB link[/URL]

[QUOTE=GP2;536354] fully-factored or probably-fully-factored <snip>
The cofactor has already been certified prime.
[/QUOTE]
Then, it is not "probable", but sure :)
Congrats to the finder and all contributors!

[QUOTE=LaurV;536368]Then, it is not "probable", but sure :)[/QUOTE]
The set, of which it is the 343rd, consists of fully-factored and probably-fully-factored numbers.

[QUOTE=LaurV;536368]Then, it is not "probable", but sure :)
Congrats to the finder and all contributors![/QUOTE]OR.

Not EXOR

[QUOTE=axn;536372]fully-factored [U][B]and[/B][/U] probably-fully-factored numbers.[/QUOTE]

[QUOTE=xilman;536376]OR.
Not EXOR[/QUOTE]
Yaarrr :chappy:
Ye nitpickers!

:rofl:

Least unproven cofactor
 

What is the least unproven Mersenne cofactor? I would like to ECPP it.

NVM. I found M78,737 on [url]https://www.mersenne.ca/prp.php[/url]

I did a Lucas PRP test with PFGW on the 42 PRPs here: [url]https://www.mersenne.ca/prp.php[/url]

Not surprisingly they are all Lucas PRP:

[CODE]Primality testing (2^78737-1)/23714605956035916529/67059801476528402969297162417 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 78577-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^78737-1)/23714605956035916529/67059801476528402969297162417 is Lucas PRP! (58.6418s+0.0154s)

Primality testing (2^82939-1)/867140681119/1018662740943783967 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 82840-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^82939-1)/867140681119/1018662740943783967 is Lucas PRP! (55.6972s+0.0006s)

Primality testing (2^84211-1)/1347377/31358793176711980763958121/3314641676042347824169591561 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 84015-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.05%
(2^84211-1)/1347377/31358793176711980763958121/3314641676042347824169591561 is Lucas PRP! (51.4970s+0.0005s)

Primality testing (2^86137-1)/2584111/7747937967916174363624460881 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 11, base 11+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 86024-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.04%
(2^86137-1)/2584111/7747937967916174363624460881 is Lucas PRP! (53.4370s+0.0007s)

Primality testing (2^86371-1)/41681512921035887 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 86316-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^86371-1)/41681512921035887 is Lucas PRP! (70.6683s+0.0006s)

Primality testing (2^87691-1)/500982892169/1610747697738457 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 11, base 1+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 8K, Pass1=128, Pass2=64, clm=2 on A 87602-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^87691-1)/500982892169/1610747697738457 is Lucas PRP! (69.0134s+0.0006s)

Primality testing (2^106391-1)/286105171290931103 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 10K, Pass1=128, Pass2=80, clm=2 on A 106334-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^106391-1)/286105171290931103 is Lucas PRP! (108.1303s+0.0006s)

Primality testing (2^130439-1)/260879 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 130422-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^130439-1)/260879 is Lucas PRP! (122.1655s+0.0006s)

Primality testing (2^136883-1)/536581361 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 136855-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.03%
(2^136883-1)/536581361 is Lucas PRP! (140.2193s+0.0006s)

Primality testing (2^151013-1)/61157791169561859593299975690769 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 13, base 1+sqrt(13)
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 150908-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.03%
(2^151013-1)/61157791169561859593299975690769 is Lucas PRP! (163.7675s+0.0006s)

Primality testing (2^157457-1)/4612545359/358012521626153 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 4+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 15K, Pass1=320, Pass2=48, clm=2 on A 157377-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^157457-1)/4612545359/358012521626153 is Lucas PRP! (205.8268s+0.0016s)

Primality testing (2^173867-1)/52536637502689 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 18K, Pass1=384, Pass2=48, clm=2 on A 173822-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^173867-1)/52536637502689 is Lucas PRP! (207.3193s+0.0007s)

Primality testing (2^174533-1)/193594572654550537/91917886778031629891960890057 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 18K, Pass1=384, Pass2=48, clm=2 on A 174380-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^174533-1)/193594572654550537/91917886778031629891960890057 is Lucas PRP! (282.0728s+0.0006s)

Primality testing (2^175631-1)/92733169/330463093135534238072561 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 18K, Pass1=384, Pass2=48, clm=2 on A 175527-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^175631-1)/92733169/330463093135534238072561 is Lucas PRP! (229.2851s+0.0053s)

Primality testing (2^216317-1)/9551099878153/42354904941257/1528559546583299567/6527839497610595205744558551 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 21K, Pass1=448, Pass2=48, clm=2 on A 216076-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^216317-1)/9551099878153/42354904941257/1528559546583299567/6527839497610595205744558551 is Lucas PRP! (318.0994s+0.0096s)

Primality testing (2^221509-1)/292391881 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 21K, Pass1=448, Pass2=48, clm=2 on A 221481-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^221509-1)/292391881 is Lucas PRP! (378.0034s+0.0006s)

Primality testing (2^270059-1)/540119/6481417/7124976157756725967 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 28K, Pass1=448, Pass2=64, clm=2 on A 269955-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^270059-1)/540119/6481417/7124976157756725967 is Lucas PRP! (545.4598s+0.0007s)

Primality testing (2^271211-1)/613961495159 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 28K, Pass1=448, Pass2=64, clm=2 on A 271172-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^271211-1)/613961495159 is Lucas PRP! (532.5829s+0.0012s)

Primality testing (2^271549-1)/238749682487 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 28K, Pass1=448, Pass2=64, clm=2 on A 271512-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.03%
(2^271549-1)/238749682487 is Lucas PRP! (544.3375s+0.0006s)

Primality testing (2^406583-1)/813167 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 40K, Pass1=640, Pass2=64, clm=2 on A 406564-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^406583-1)/813167 is Lucas PRP! (1213.9665s+0.0006s)

Primality testing (2^432457-1)/1672739247834685086279697 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 48K, Pass1=768, Pass2=64, clm=2 on A 432377-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.02%
(2^432457-1)/1672739247834685086279697 is Lucas PRP! (1514.2660s+0.0007s)

Primality testing (2^440399-1)/880799/31518475633/16210820281161978209 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 6+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 48K, Pass1=768, Pass2=64, clm=2 on A 440281-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^440399-1)/880799/31518475633/16210820281161978209 is Lucas PRP! (1616.8335s+0.0009s)

Primality testing (2^488441-1)/61543567/30051203516986199 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 48K, Pass1=768, Pass2=64, clm=2 on A 488361-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^488441-1)/61543567/30051203516986199 is Lucas PRP! (1886.7971s+0.0007s)

Primality testing (2^576551-1)/4612409/64758208321/242584327930759 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 60K, Pass1=768, Pass2=80, clm=2 on A 576446-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^576551-1)/4612409/64758208321/242584327930759 is Lucas PRP! (2427.9374s+0.0085s)

Primality testing (2^611999-1)/18464214225958267477777390354183 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 11, base 8+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 60K, Pass1=768, Pass2=80, clm=2 on A 611896-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^611999-1)/18464214225958267477777390354183 is Lucas PRP! (2210.3935s+0.0012s)

Primality testing (2^675977-1)/1686378749257/7171117283326998925471 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 4+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 72K, Pass1=384, Pass2=192, clm=4 on A 675864-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^675977-1)/1686378749257/7171117283326998925471 is Lucas PRP! (3419.3362s+0.0007s)

Primality testing (2^684127-1)/23765203727 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 72K, Pass1=384, Pass2=192, clm=4 on A 684093-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^684127-1)/23765203727 is Lucas PRP! (3200.8319s+0.0044s)

Primality testing (2^696343-1)/11141489/36009913139329 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 72K, Pass1=384, Pass2=192, clm=4 on A 696275-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.01%
(2^696343-1)/11141489/36009913139329 is Lucas PRP! (3238.8290s+0.0046s)

Primality testing (2^750151-1)/429934042631/7590093831289/397764574647511/8361437834787151/17383638888678527263 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 72K, Pass1=384, Pass2=192, clm=4 on A 749905-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^750151-1)/429934042631/7590093831289/397764574647511/8361437834787151/17383638888678527263 is Lucas PRP! (4287.7923s+0.0043s)

Primality testing (2^822971-1)/6583769/28211445881/21255852651726486149207 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 80K, Pass1=320, Pass2=256, clm=4 on A 822840-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^822971-1)/6583769/28211445881/21255852651726486149207 is Lucas PRP! (5668.1033s+0.0034s)

Primality testing (2^1010623-1)/12602017578957977 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 100K, Pass1=320, Pass2=320, clm=4 on A 1010570-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^1010623-1)/12602017578957977 is Lucas PRP! (6651.6134s+0.0039s)

Primality testing (2^1168183-1)/54763676838381762583 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 7+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 112K, Pass1=448, Pass2=256, clm=2 on A 1168118-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^1168183-1)/54763676838381762583 is Lucas PRP! (11847.6134s+0.0027s)

Primality testing (2^1304983-1)/52199321 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 128K, Pass1=512, Pass2=256, clm=2 on A 1304958-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^1304983-1)/52199321 is Lucas PRP! (11420.8682s+0.0034s)

Primality testing (2^1629469-1)/644908484660139264379223 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 160K, Pass1=640, Pass2=256, clm=2 on A 1629390-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^1629469-1)/644908484660139264379223 is Lucas PRP! (25361.2088s+0.0496s)

Primality testing (2^1790743-1)/146840927/158358984977/3835546416767873/20752172271489035681 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 13, base 10+sqrt(13)
Generic modular reduction using generic reduction FMA3 FFT length 192K, Pass1=768, Pass2=256, clm=2 on A 1790563-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^1790743-1)/146840927/158358984977/3835546416767873/20752172271489035681 is Lucas PRP! (26036.0751s+0.0050s)

Primality testing (2^2327417-1)/23915387348002001 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Generic modular reduction using generic reduction FMA3 FFT length 240K, Pass1=320, Pass2=768, clm=1 on A 2327363-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^2327417-1)/23915387348002001 is Lucas PRP! (38071.2133s+0.0098s)

Primality testing (2^3464473-1)/604874508299177 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 384K, Pass1=384, Pass2=1K, clm=4 on A 3464424-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^3464473-1)/604874508299177 is Lucas PRP! (107647.0818s+0.0132s)

Primality testing (2^4187251-1)/72234342371519 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 448K, Pass1=448, Pass2=1K, clm=4 on A 4187205-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^4187251-1)/72234342371519 is Lucas PRP! (159196.3674s+0.0201s)

Primality testing (2^4834891-1)/1701881633/70659688575577 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 480K, Pass1=384, Pass2=1280, clm=4 on A 4834815-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^4834891-1)/1701881633/70659688575577 is Lucas PRP! (240488.3063s+0.0189s)

Primality testing (2^5240707-1)/75392810903 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Generic modular reduction using generic reduction FMA3 FFT length 560K, Pass1=448, Pass2=1280, clm=4 on A 5240671-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^5240707-1)/75392810903 is Lucas PRP! (260942.1577s+0.0389s)

Primality testing (2^7080247-1)/156822217506727/11283326312536321/9632940548330339593 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 5+sqrt(5)
Generic modular reduction using generic reduction FMA3 FFT length 720K, Pass1=320, Pass2=2304, clm=4 on A 7080084-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^7080247-1)/156822217506727/11283326312536321/9632940548330339593 is Lucas PRP! (493240.3769s+0.0461s)

Primality testing (2^7313983-1)/305492080276193 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 11, base 1+sqrt(11)
Generic modular reduction using generic reduction FMA3 FFT length 768K, Pass1=384, Pass2=2K, clm=2 on A 7313935-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 0.00%
(2^7313983-1)/305492080276193 is Lucas PRP! (387269.4697s+0.0243s)

[/CODE]

[M]M10,443,557[/M] has been reported fully factored.

This would be a new record. The previous record-holder was [M]M7,313,983[/M]

[QUOTE=GP2;550090][M]M10,443,557[/M] has been reported fully factored.

This would be a new record. The previous record-holder was [M]M7,313,983[/M][/QUOTE]

Congrats! I'll leave it to ATH to do a Lucas test.

[QUOTE=paulunderwood;550093]Congrats! I'll leave it to ATH to do a Lucas test.[/QUOTE]

You are welcome to run it with your own software. PFGW takes a very long time because it can only use 1 core and because it does a lot of other tests before starting the Lucas test. Last 2 took 4.5 and 5.7 days, this would probably take 8+ days.

[QUOTE=ATH;550103]You are welcome to run it with your own software. PFGW takes a very long time because it can only use 1 core and because it does a lot of other tests before starting the Lucas test. Last 2 took 4.5 and 5.7 days, this would probably take 8+ days.[/QUOTE]

Will do...