M332263333 has a factor: 7564203664111030934537
Assigned LL testing to "ANONYMOUS" on 2011-05-27
:shark:

nice one uncwilly

nice find! I hope that anonymous guy read the forum from time to time! Or at least I hope he did not start it, so P95 will detect the invalid key and cancel it, saving the guys a lot of time, computing power and energy (P95 does not cancel an assignment if work was already done on it).

P-1 found a factor in stage #2, B1=550000, B2=10312500.
UID: Jwb52z/Clay, M59112883 has a factor: 3888937701898611502946084809

91.651 bits.

(M1061)+1 has factor: 2

[COLOR=White]Sorry, boredom got a hold of me :/[/COLOR]

has it any other prime factor?

[QUOTE=firejuggler;319431]has it any other prime factor?[/QUOTE]
Yes. There are 1060 more factors of 2 that can be taken :)
If you mean other than 2, I'm afraid that's a negative. :P

P-1 found a factor in stage #1, B1=550000.
UID: Jwb52z/Clay, M59224261 has a factor: 40037708579582423922112437113

95.015 bits.

I've done a little bit of work on the low 400s, since the entire range seems to be neglected...

[code] Manual testing 400211897 78486019961985822737
Manual testing 400211531 70083357476876969089
Manual testing 400211401 80312150767930776041
Manual testing 400210667 254779352043238285561
Manual testing 400210561 136205572610484789329
Manual testing 400210267 46457502502910894713
Manual testing 400210193 202660594529809645153
Manual testing 400209539 117886080274723663223
Manual testing 400209079 124216227029209499927
Manual testing 400208087 54014399967601218799
Manual testing 400207573 148672114308670369591
Manual testing 400207391 277885690908888134713
Manual testing 400207001 139862133803757024593
Manual testing 400206277 46560008834026950463
Manual testing 400205669 166791254458979859823
Manual testing 400202617 162354277660260706897
Manual testing 400202203 73078758015315271151
Manual testing 400201951 86027759777165615737
Manual testing 400999831 64377936036388210889
Manual testing 400998943 56004253858505522441
Manual testing 400998799 44647762122739241057
Manual testing 400996367 64370756261960144489
Manual testing 400989769 61199751342021717143
Manual testing 400982773 53654918980804300793
Manual testing 400981951 44943135922048624031
Manual testing 400975723 54163206650988336233
Manual testing 400972753 61748610448957894369
Manual testing 400971941 43522050023566199383
Manual testing 400959971 67950333729286699121
Manual testing 400955173 63796965969023141609
Manual testing 400953919 64882164504724582529

[/code]

P-1 found a factor in stage #1, B1=550000.
UID: Jwb52z/Clay, M59219323 has a factor: 4003973722867683284562263

81.728 bits.

I found another 40-digit prime factor of a Mersenne number using P-1 algorithm:

[Sun Nov 25 02:01:52 2012]
P-1 found a factor in stage #2, B1=100000000, B2=2000000000.
M102679 has a factor: 2744975274947255820412194963218834106959

k = 71 × 379 × 587 × 49613 × 3386321 × 9968291 × 505297129

The biggest factor I have is
18164039969328095522036298121273856953
Exponent=[URL="http://www.mersenne.org/report_exponent/?exp_lo=53324207"]53324207[/URL]
Bit Level=123.77

Yeah, P-1 of course :smile:
...Found quite a while ago though, heh by my laptop.

[QUOTE=kracker;319601]The biggest factor I have is
18164039969328095522036298121273856953
Exponent=[URL="http://www.mersenne.org/report_exponent/?exp_lo=53324207"]53324207[/URL]
Bit Level=123.77

Yeah, P-1 of course :smile:
...Found quite a while ago though, heh by my laptop.[/QUOTE]

[URL="http://www.mersenne.ca/exponent.php?factordetails=18164039969328095522036298121273856953"]S1 too[/URL]! Nice find!

I found a TF factor in less than 13 minutes (wish they were all that fast):
[CODE]
got assignment: exp=[URL="http://www.mersenne.ca/exponent.php?exponentdetails=60330379"]60330379[/URL] bit_min=70 bit_max=73 (27.75 GHz-days)
Starting trial factoring M60330379 from 2^70 to 2^73 (27.75 GHz-days)
k_min = 9784387568340
k_max = 78275100557045
Using GPU kernel "barrett76_mul32"
class | candidates | time | ETA | avg. rate | SievePrimes | CPU wait
96/4620 | 3.57G | 34.979s | 9h06m | 101.95M/s | 7434 | 2.25%
M60330379 has a factor: 7796005832769103194769
found 1 factor for M60330379 from 2^70 to 2^73 (partially tested) [mfaktc 0.19 barrett76_mul32]
tf(): total time spent: [B]12m 59.980s[/B]
[/CODE]

M33981841 has a factor: 443174522098036005337 [TF:68:69*:mfakto 0.12-Win barrett15_75]
M33969223 has a factor: 303371866484685283463 [TF:68:69*:mfakto 0.12-Win barrett15_75]
M33570463 has a factor: 371149646254495501577 [TF:68:69*:mfakto 0.12-Win barrett15_75]

Kracker, the * mean partially tested, right?

[QUOTE=firejuggler;319634]Kracker, the * mean partially tested, right?[/QUOTE]

Yeah I think so, should I fully test it?

I don't think they need full test, noone will bother with these exponents in the future. For such high exponents and bitlevels, once a factor is found the exponent is forgotten. I don't full test them, anyhow, when I found a factor.

And the 0.7% speedup is worth missing a potential second factor? I don't think so. :no:

Ah, I see.

M33554501 has a factor: 474399666665717488391 [TF:68:69*:mfakto 0.12-Win barrett15_75]
M33591083 has a factor: 319291795341306602569 [TF:68:69*:mfakto 0.12-Win barrett15_75]
M33824533 has a factor: 323303457724858544951 [TF:68:69*:mfakto 0.12-Win barrett15_75]

3 today so far, heh more than usual.

Nice "stage-2" found, not terrible high, but remarkable because TF missed it by only a few couples of prime candidates:

M59168051 F-PM1 12582068973364790994337
only 73 bits and a [SUB]little bit[/SUB]... :razz:

k=2^4*3*11*19*2767*[B]3830347[/B]
the B1/B2 used (default P95 values): 555000/11238750, so not even a difficult one... [SIZE=2][/SIZE]

M64312097 has a factor: 1193158778036005925513 70 bits.

A lucky week, four factors for small exponents, all found by P-1 with B1=25000000, B2=1000000000. Nothing remarkable about any of them, apart from one being found in stage 1 which I don't have often in this range.

M511171 has a factor: 266998244095654643023390393393, 98 bits
M511439 has a factor: 2186799313429158965622453983, 91 bits
M511633 has a factor: 50392350975337586383793, 76 bits
M512507 has a factor: 22486400320827680813033, 75 bits

And as a bonus, one P-1 factor up in the active ranges.
M59264501 has a factor: 2653474548638029644824309417, 92 bits

M59014499
 

Hadn't had a P-1 success for a while, until yesterday evening: 2072093725551089765095534553 is a factor of M59014499. 90.743 bits and prime. [TEX]k = 2^2 \times 37 \times 383 \times 4397 \times 115361 \times 610583[/TEX]. This was found in Stage 2, but just beyond the B1-limit of 550,000. The B2-limit was 10,725,000, about 17-1/2 times what would have been needed.

P-1 found a factor in stage #2, B1=555000, B2=10406250.
UID: Jwb52z/Clay, M59636569 has a factor: 142805156199906713570257

76.918 bits.

A couple of factors in the 334M range.

P-1 found a factor in stage #1, B1=555000.
UID: Jwb52z/Clay, M59835911 has a factor: 75271232967983331326401

75.995 bits.

P-1 found a factor in stage #2, B1=560000, B2=10640000.
UID: Jwb52z/Clay, M60030521 has a factor: 56376288867390220894969

75.758 bits.

P-1 found a factor in stage #1, B1=560000.
UID: Jwb52z/Clay, M60074737 has a factor: 277595962638024974862031

77.877 bits.

P-1 found a factor in stage #2, B1=560000, B2=10640000.
UID: Jwb52z/Clay, M60165101 has a factor: 16306920507796446029681

73.788 bits.

P-1 found a factor in stage #2, B1=560000, B2=10640000.
UID: Jwb52z/Clay, M60181259 has a factor: 150985799039081099694123973633

96.930 bits.

Hi,

found a [B]composite factor[/B] in [B]stage #2[/B] this week.

P-1 found a factor in stage #2, [B]B1=565000[/B], B2=11723750, E=12.
M59989133 has a factor: 10498651208818561297204833414014330244934487388503659231 (182.8 bits)

F[SUB]1[/SUB] = 82258643608965470493798023 (86.1 bits)
F[SUB]2[/SUB] = 127629763246841337664420837097 (96.7 bits)

k[SUB]1[/SUB] = 71 * 64187 * 161221 * [B]933151[/B]
k[SUB]2[/SUB] = 2[SUP]2[/SUP] * 7019 * 37589 * 245789 * [B]4101011[/B]

Oliver

[QUOTE=TheJudger;323876]Hi,

found a [B]composite factor[/B] in [B]stage #2[/B] this week.

P-1 found a factor in stage #2, [B]B1=565000[/B], B2=11723750, E=12.
M59989133 has a factor: 10498651208818561297204833414014330244934487388503659231 (182.8 bits)

F[SUB]1[/SUB] = 82258643608965470493798023 (86.1 bits)
F[SUB]2[/SUB] = 127629763246841337664420837097 (96.7 bits)

k[SUB]1[/SUB] = 71 * 64187 * 161221 * [B]933151[/B]
k[SUB]2[/SUB] = 2[SUP]2[/SUP] * 7019 * 37589 * 245789 * [B]4101011[/B]

Oliver[/QUOTE]

:huh2::bounce wave:

Oh my... Nice.

[QUOTE=TheJudger;323876]
k[SUB]1[/SUB] = 71 * 64187 * 161221 * [B]933151[/B]
k[SUB]2[/SUB] = 2[SUP]2[/SUP] * 7019 * 37589 * 245789 * [B]4101011[/B]
[/QUOTE]

Hm... This looks like a Brent Suyama hit! (grrr, I was first writing "BS hit" and then realized how that sounds!). Both tailing terms are higher than B1, and their combined product is higher than B2. You can not find the combined (composite) factor with normal B1<933k, B2<5M. You will find them one by one, but not their product, unless P95 computes all the primes between the two values in a gulp. I saw P95 computing 2400 primes in a time for lower exponents, but for this high, no way. Regardless of the fact that there are 216141 primes in between the two values, there is no way P95 gulped all those in a single bite (between two GCD's) in stage 2. [edit: or... is it?]

P-1 found a factor in stage #2, B1=25000000, B2=1000000000, E=12.
M428899 has a factor: 125121243653506108640827021803878584977256889
45 decimal digits, 147 bits
k = 2^2 * 73 * 83 * 3571 * 3923 * 12583 * 13513 * 37691 * 84307 * 795132769

My first reaction was "nice, a composite factor". I was a bit stunned when it wasn't.

[QUOTE=markr;324131]P-1 found a factor in stage #2, B1=25000000, B2=1000000000, E=12.
M428899 has a factor: 125121243653506108640827021803878584977256889
45 decimal digits, 147 bits
k = 2^2 * 73 * 83 * 3571 * 3923 * 12583 * 13513 * 37691 * 84307 * 795132769

My first reaction was "nice, a composite factor". I was a bit stunned when it wasn't.[/QUOTE]

Not the record for largest B2(no B-S)/B2(actual), but that's still massive. :smile:

James, is there a simple "largest P-1 factors" list on your site?

[QUOTE=Dubslow;324142]Not the record for largest B2(no B-S)/B2(actual), but that's still massive. :smile:

James, is there a simple "largest P-1 factors" list on your site?[/QUOTE]

Here you go:
[url]http://mersenne.ca/stats.php?showuserstats=*[/url]
FWIW, it would take 3*10^25 TF Ghz-days to find that factor!

In that page it appears that I found a prime factor of M102679 twice, using P-1 and ECM methods. This is a known problem when copying results.txt in the Manual results page. Primenet says that it was found by ECM, while I used P-1. Then I opened the [url]http://mersenne.ca[/url] Web site and dumped my results.txt file there. Now the prime factor appears twice in this Web site.

P-1 found a factor in stage #2, B1=560000, B2=10640000.
UID: Jwb52z/Clay, M60465473 has a factor: 12850552559520705957651673

83.410 bits.

P-1 found a factor in stage #1, B1=560000.
UID: Jwb52z/Clay, M60381703 has a factor: 251637327815644828195129

77.736 bits

P-1 found a factor in stage #2, B1=570000, B2=10687500.
UID: Jwb52z/Clay, M60775031 has a factor: 434496528948485397527849

78.524 bits.

I had some fun this morning when I just saw this, but I had to run so the only thing I had time to do is to approximate the bit level... >170 bits.:fusion:
Then I had to run off to work...

[QUOTE][Sat Feb 09 06:01:04 2013]
P-1 found a factor in stage #2, B1=560000, B2=10640000.
UID: Axelsson/Utveckling, M60070949 has a factor: 91622368695369671284251815722954195372651812038785409
[/QUOTE]almost eight hours later I had time to report my find... what a disappointment, i had found a composite factor. It was just too good to be true and even though I had felt that something was wrong, reality sucks! :ouch2:
Anyhow, my first composite factor from the P-1 method, how common is it with composite factors in P-1?

[QUOTE]Factors found: 1 Processing result: M60070949 has a factor: 91622368695369671284251815722954195372651812038785409 Composite factor 91622368695369671284251815722954195372651812038785409 = 1175315009936832799493321 * 77955584605606208779422907129[/QUOTE]79.96 bit and 95.98 bit, large but not extreme.

My next goal, a triple factor... :big grin:

/Göran

The largest 'non composite' factor found with P-1 are around 100 bits. A 170 bit composite is still a medal contender.
additionnal info
k1(1175315009936832799493321 )= 2^2 * 5 *491 * 377297 * 2640371
k2(77955584605606208779422907129)= 2^2 * 3 * 3719 * 102149 * 115553 * 1231771

meaning *optimal * bound would have been
- if you looked for a stage 1 hit : B1=2640371 B2=2640371
- if you looked for a stage 2 hit : B1=377297 B2=2640371
That would have found the 2 factor and would be faster.
But since we didn't know the factor beforehand, good job.

[QUOTE=firejuggler;328711]The largest 'non composite' factor found with P-1 are around 100 bits. A 170 bit composite is still a medal contender.[/QUOTE]

My largest was 104 bits. I've heard of people finding larger, but they are rare.

Composite factors are rare, but not as rare as 140+ bit prime factors. Any factor with more bits than twice the factored depth is almost certainly composite.

[QUOTE=Mr. P-1;328715]My largest was 104 bits. I've heard of people finding larger, but they are rare.[/QUOTE]There is a 102 bit number in the 100M digit range. M332219539 has 6157103818187609775710294086831 as a factor.

[URL="http://mersenneforum.org/showpost.php?p=319601&postcount=594"]..[/URL]is my largest.

I'm not sure that this is the right place for this, given that the exponent is composite, but...

M1369 has a factor: 6337549872222510181433065176903666450905719.

The cofactor, (after dividing out this and the previously known algebraic and non-algebraic factors) is a PRP349, which completes the factorisation of this exponent.

Found by ecm stage 2. sigma=2257798857, B1=43000000, B2=default.

Group Order: 2^3 · 3^2 · 17 · 457 · 15901 · 44449 · 88897 · 543611 · 6754669 · 49108642183

Before I found the factor, I had been thinking that I would probably not see the exponent fully factored in my lifetime. My reasoning was that it was about a billion times harder than the largest SNFS job so far completed. (I'm 48 years old. I don't have time for Moore's law to catch up.) The exponent was low, so probably already had been subject to a lot of ECM, and even finding one or two small factors would not reduce the GNFS difficulty below the SNFS difficulty.

I concluded that my best hope was to find a small factor and discover that the cofactor was prime. I did not think this was likely. Nevertheless I was prepared to take the number up to t50, which I figured I could do in about a month or less using one core. In the end, it only took a few hours.

So, this time , luck was the determinant factor.

Wowzers. That was incredibly lucky!

[QUOTE=Mr. P-1;328837]In the end, it only took a few hours.[/QUOTE]

Wish they were all like that!

Woohaaa! Congrats! Nice finding.

[QUOTE=Mr. P-1;328837]I'm not sure that this is the right place for this, given that the exponent is composite, but...

M1369 has a factor: 6337549872222510181433065176903666450905719.

The cofactor, (after dividing out this and the previously known algebraic and non-algebraic factors) is a PRP349, which completes the factorisation of this exponent.

Found by ecm stage 2. sigma=2257798857, B1=43000000, B2=default.

Group Order: 2^3 · 3^2 · 17 · 457 · 15901 · 44449 · 88897 · 543611 · 6754669 · 49108642183
[/QUOTE]
[EMAIL="ssw@cerias.purdue.edu"]Send e-mail to Sam Wagstaff[/EMAIL] ; he has [URL="http://homes.cerias.purdue.edu/~ssw/cun/xtend/index.html"]extension[/URL] pages.
Congrats!

[QUOTE=flashjh;328895]Wish they were all like that![/QUOTE]

If they were, then they would all have been found by now.

[QUOTE=Dubslow;328877]Wowzers. That was incredibly lucky![/QUOTE]

Lucky in what sense? Lucky that there was a P43 waiting to be found? That depends upon how much ECM other people had done before me. If, as I had assumed, it was "a lot", then, then it's more a case that these other searchers had been unlucky. On the other hand, my success might suggest that there is still a lot of low-hanging fruit among the composite-exponent near-Cunninghams. (Perhaps also the prime-exponents other than 2-.)

If you meant lucky to find the factor so quickly, the expected time to t40 was about a day or so for a single core on my machine. I found the P43 in a few hours. Lucky, but not incredibly so.

[QUOTE=Batalov;328918][EMAIL="ssw@cerias.purdue.edu"]Send e-mail to Sam Wagstaff[/EMAIL] ; he has [URL="http://homes.cerias.purdue.edu/~ssw/cun/xtend/index.html"]extension[/URL] pages.
Congrats![/QUOTE]

Sent. Also to Will Edgingon.

[QUOTE=Mr. P-1;328837]The cofactor, (after dividing out this and the previously known algebraic and non-algebraic factors) is a PRP349[/QUOTE]

Now [url=http://factordb.com/index.php?id=1100000000584435267]confirmed[/url] as a P.

P-1 found a factor in stage #2, B1=580000, B2=10730000.
UID: Jwb52z/Clay, M61367717 has a factor: 81843081338400211462447

76.115 bits

M89576251 has a factor: 391226059433524759463 (68.4 bits)
k = 701 * 3,115,206,781.

[QUOTE=Mr. P-1;328963]Lucky in what sense? Lucky that there was a P43 waiting to be found? That depends upon how much ECM other people had done before me.[/QUOTE]
You have a very nice find, but I don't think it is that rare. For example, I found a 42-digit factor of the c198 of [URL="http://www.factordb.com/index.php?id=1000000000012151257"]2^1233+1[/URL] on January 2, this year, with B1=3000000. While I can't be certain I was the first to find it, it wasn't in factordb. [There is still a c157, and I have done "just enough" ECM that it's a statistical tie whether ECM or GNFS is more efficient to finish it. (I moved the c157 to my GNFS queue, but it will be months before I get to it, if ever. I do not claim any sort of "reservation" on this number.)]

As you say, it all depends how much ECM other people have done before us. Once you get above the Cunningham Table limits (including proposed extensions), there seem to be a lot of numbers that haven't been worked extensively.

[QUOTE=rcv;328990]While I can't be certain I was the first to find it, it wasn't in factordb.[/QUOTE]

Obviously I can't be certain about mine either, but I'd checked [url=http://www.garlic.com/~wedgingt/mersenne.html]Edgington's database[/url] as well as factordb. Unfortunately I'd missed [url=http://homes.cerias.purdue.edu/~ssw/cun/xtend/other]this[/url] and [url=http://homes.cerias.purdue.edu/~ssw/cun/xtend/crombie]this[/url]. Fortunately my find wasn't on either list. It's there now.

new factor found
 

First time I ever messed with GPU/mfaktc, (my) first factor..beginner's luck...

M83833487 has a factor: 318129387684576210031 (68.1 bits)
k=3×5×23×431×12760271

M60809621 has a factor: 2905866497364711208590790240514270083169
131.09 Bits; k = 23893147577459093262485472821104 = 2 * 2 * 2 * 2 * 271 * 1451 * 3037 * 13441 * 210601 * 329209 * 1341863

Slightly below my personal record, I guess it is my 2nd largest so far.

P-1 found a factor in stage #1, B1=580000.
UID: Jwb52z/Clay, M61705447 has a factor: 2614309415709170708616361

81.113 bits.

My first P-1 success with my new box:

[code][Thu Feb 21 23:47:32 2013]
P-1 found a factor in stage #2, B1=585000, B2=11992500, E=6.
UID: daran/agogo, M61797409 has a factor: 13497681001092947908111, AID: 8941DF002A23E43404FBEEE76A6CA876[/code]

74 bits (previously TFed to 73). k=3*5*109*27827*800117.

P-1 factor for [COLOR=black]59,750,861[/COLOR], [URL="http://www.mersenne.ca/factor/87542740676833513855994017"]87542740676833513855994017[/URL] 86 bits.

Nuts. My first P-1 factor after rejoining the effort was a paltry 74.09 bits. The only reason it even merits a post is the ridiculously tiny/smooth k: [URL="http://www.mersenne.ca/exponent.php?exponentdetails=59898271"]min B2 is only 4751[/URL].

Looks like you got something a [strike]LOT[/strike] bit better!

77.8 bits for you! :razz:

[COLOR=Silver]Hmm, also it looks like [strike]I found[/strike] minion No. 7 found a P-1 factor just now, [SIZE=2]542430480860023763539783 for [/SIZE][/COLOR][SIZE=2][COLOR=Silver]59928367[/COLOR][SIZE=2][COLOR=Silver], 78 bits.[/COLOR]
[/SIZE][/SIZE]

Actually, that computer got very lucky overnight, but still paltry factors of 74.2 and 77.9 (again) bits. The 597049xx range has been quite lucrative for P-1 factors.

Edit: It seems kracker just got another hit of his own, also above 78 bits :razz:

[code]kracker ccc-node-07 59928367 F-PM1 2013-02-25 20:09 0.0 542430480860023763539783[/code]

P-1 found a factor in stage #2, B1=580000, B2=10730000.
UID: Jwb52z/Clay, M61769167 has a factor: 3552790399080178821438689

81.555 bits.

Here's a good one: 12395067022025302245408215823361, 104 bits.
(from which you can see that not all prime exponents containing the "number of the best" will generate prime mersennes :razz:)

[QUOTE=LaurV;331033]Here's a good one: 12395067022025302245408215823361, 104 bits.
(from which you can see that not all prime exponents containing the [STRIKE]"number of the best"[/STRIKE] will generate prime mersennes :razz:)[/QUOTE]
"number of the BEAST"

[QUOTE=c10ck3r;331038]"number of the BEAST"[/QUOTE]
grrr, typo, thanks

M95516249819281897377329792747204747051937323318703014161875271870743369159415765467622402105668679795781959994061775011707821819120345026036123142898041417744024085438782757861002649408975294945304266551957165557916259132607823871 has a factor: 191032499638563794754659585494409494103874646637406028323750543741486738318831530935244804211337359591563919988123550023415643638240690052072246285796082835488048170877565515722005298817950589890608533103914331115832518265215647743

[QUOTE=jnml;331225]M95516249819281897377329792747204747051937323318703014161875271870
7433691594157654676224021056686797957819599940617750117078218191203450260361231428980
41417744024085438782757861002649408975294945304266551957165557916259132607823871 has a factor: 191032499638563794754659585494409494103874646637406028323750543741486738318831530935244
804211337359591563919988123550023415643638240690052072246285796082835488048170877565515
722005298817950589890608533103914331115832518265215647743[/QUOTE]

I'm just curious. How did you get these two numbers? And what's special of the exponent?

[QUOTE=dabaichi;331245]I'm just curious. How did you get these two numbers? And what's special of the exponent?[/QUOTE]

Nothing special above the fact that both the factor and the exponent are PRPs. Otherwise it's a 2p+1 | Mp (i.e. k == 1), so I actually cheated (in a sense).

Got that from a toy program of mine, it's just the size surprised me a bit ;-)

[CODE]
(16:38) jnml@fsc-r550:~/src/tmp/mersenne/ff$ time ./ff
2: 23 | M11 (5 bits)
2: 47 | M23 (6 bits)
2: 383 | M191 (9 bits)
14: 479 | M239 (9 bits)
20: 167 | M83 (8 bits)
26: 863 | M431 (10 bits)
26: 7421703487487 | M3710851743743 (43 bits)
26: 143556623544770914273611162510277214207 | M71778311772385457136805581255138607103 (127 bits)
32: 263 | M131 (9 bits)
32: 290271069732863 | M145135534866431 (49 bits)
44: 359 | M179 (9 bits)
44: 719 | M359 (10 bits)
44: 1439 | M719 (11 bits)
44: 2879 | M1439 (12 bits)
44: 23039 | M11519 (15 bits)
44: 1474559 | M737279 (21 bits)
44: 2949119 | M1474559 (22 bits)
44: 24159191039 | M12079595519 (35 bits)
44: 6184752906239 | M3092376453119 (43 bits)
50: 52223 | M26111 (16 bits)
56: 1823 | M911 (11 bits)
56: 3825205247 | M1912602623 (32 bits)
62: 503 | M251 (9 bits)
62: 191032499638563794754659585494409494103874646637406028323750543741486738318831530935244804211337359591563919988123550023415643638240690052072246285796082835488048170877565515722005298817950589890608533103914331115832518265215647743 | M95516249819281897377329792747204747051937323318703014161875271870743369159415765467622402105668679795781959994061775011707821819120345026036123142898041417744024085438782757861002649408975294945304266551957165557916259132607823871 (765 bits)
^C
real 0m31.379s
user 0m31.070s
sys 0m0.080s
(16:38) jnml@fsc-r550:~/src/tmp/mersenne/ff$
[/CODE]

EDIT: The code is now available [URL="https://github.com/cznic/mathutil/blob/master/ff/main.go"]here[/URL].

factor5 confirm the factor found.

[QUOTE=firejuggler;331273]factor5 confirm the factor found.[/QUOTE]
Thanks for the confirmation. Are such "factors" worth of reporting here? For example:

[CODE]
jnml@fsc-r630:~/src/github.com/cznic/mathutil/ff$ time ./ff -c 1382 -d 10s
1382: 23202889727 | M11601444863 (35 bits)
1382: 2891317468805925583122310289685036337676269648223968521576930801170224130471293604830368928083599358133453548705464010629846117567184743543864401985621482209720123399448999256896093377478118494454636064824009640486909425185943889123952719307865058813784823875371947571635800099775698380593420250938639525924694459005059051692445445662364099847064237399153239430674586254346911706508030497459384755111708766149920027758576898244058419917181406937087 | M1445658734402962791561155144842518168838134824111984260788465400585112065235646802415184464041799679066726774352732005314923058783592371771932200992810741104860061699724499628448046688739059247227318032412004820243454712592971944561976359653932529406892411937685973785817900049887849190296710125469319762962347229502529525846222722831182049923532118699576619715337293127173455853254015248729692377555854383074960013879288449122029209958590703468543 (1487 bits)
^C
real 0m6.973s
user 0m6.816s
sys 0m0.060s
jnml@fsc-r630:~/src/github.com/cznic/mathutil/ff$
[/CODE]

I'm afraid that the range considered usefull are only the M below 100 millions and the range
between 332.2 -334 millions. Outside of these range, it is not really usefull (outside of personnal quest).

[QUOTE=jnml;331251]Nothing special above the fact that both the factor and the exponent are PRPs. Otherwise it's a 2p+1 | Mp (i.e. k == 1), so I actually cheated (in a sense). ...
[/QUOTE]

Thanks for your explanations!

59933213 haz dat factor of teh 266130256367911424981339321 87.78 bitz.:ignore:

[QUOTE=jnml;331225]M9551624981928189737<snip> has a factor: 19103249963856379475<snip>[/QUOTE]... and the cofactor is ?

not prime, with a 99.9% chance.

[QUOTE=cheesehead;331602]... and the cofactor is ?[/QUOTE]
I think the cofactor is non computable (in this case).

[code][Sat Mar 2 00:40:06 2013]
P-1 found a factor in stage #1, B1=585000.
UID: daran/agogo, M61797677 has a factor: 14080954256065607739793951951487, AID: F930BE0F01061A1D5C0134A93FA87741[/code]

At 104 bits this is my largest stage 1 factor ever. k = 41 * 409 * 15683 * 25309 * 36919 * 463627.

P-1 found a factor in stage #1, B1=575000.
M60808259 has a factor: 592791747603566405334727072882584265279
128.80 Bits; k = 4874270019830418145458884728821 = 3 * 22013 * 25219 * 41999 * 391133 * 407311 * 437413
A little bit above the average size of my stage 1 results. :smile:

Oliver

Got an 88 bit hit some days ago. :smile:

[url]http://www.mersenne.ca/exponent.php?exponentdetails=61557059[/url]

I just [URL="https://www.gpu72.com/reports/worker/factors/d0a0a9610115a8227c686f0d8951998b/bydate/"]got[/URL] a 77 bit factor for (M)60181483. Boohoo. :evil:

[SIZE="1"](finding about roughly 1 P-1 factor a day)[/SIZE]

Y'know how P-1 finds factors that would take ages to find using TF? Well, does anyone have some good examples where TF would be similarly many times faster than P-1?

[QUOTE=c10ck3r;333695]Y'know how P-1 finds factors that would take ages to find using TF? Well, does anyone have some good examples where TF would be similarly many times faster than P-1?[/QUOTE]
Something with a 39- or 40-bit prime k (for the range of typical exponents now being tested) would do it, I think.

I guess, actually any k with a largest prime factor over 25 bits, which would require P-1 B2 > 30M, and a second-largest prime factor large enough to require a high B1. Something like that.

Someone could work out a more specific function depending on size of exponent and sizes of largest two prime factors of k, and post some nice curves on a graph.

(Maybe if I get energetic like when I was younger, and have spare time. Or if I'd ever become a Mathematica master like I once intended. With that 3-D display floating in the air like I dreamed up in high school calculus class, plus hand-gesture controls.

Heh.)

That's easy. For example:

Factor 8643081839402111473 (63 bits) of M60004193 would require a B2 of 142898113
Factor 6104384570413241087 (63 bits) of M60005497 would require a B2 of 3912708563
Factor 630004667994632017847 (70 bits) of M60000049 would require a B2 of 477275873857

Okay, quick question: if, for example, a prime k-value is between B1 and B2, will it be found by stage 2? For example, M1277 shows a B1 of 2*10^12 and a B2 of 3*10^13. If there was a prime k-value of about, say, 1*10^13, would stage 2 find it?
If so, does this mean it has, through P-1, been effectively TF'ed to 96.95 bits? (2*1277*2e12*3e13)

Yes to the first question. No to the second.

[QUOTE=c10ck3r;333774]M1277 shows a B1 of 2*10^12 and a B2 of 3*10^13. If there was a prime k-value of about, say, 1*10^13, would stage 2 find it?
If so, does this mean it has, through P-1, been effectively TF'ed to 96.95 bits? (2*1277*2e12*3e13)[/QUOTE]

No. P-1 will find a factor 2kp+1 in stage 2 if there is a single prime in the factorisation of k between B1 and B2, and all other primes [i]and prime powers[/i] of the factorisation of k are < B1. Theoretically k could be a prime power a little over 2*10^12 and the factor will have been missed. This means that M1277 has been "TFed" by P-1 to 2*1277*2e12 which is about 52 bits, less than the real amount of TF.

However, there has also been a considerable amount of [url=http://mersenne.org/report_ECM/]ECM[/url] done on this exponent, so much that any factor less than 60 digits (200 bits) would most likely have been found. So while it remains a theoretical possibility that a small factor remains to be found, the expectation is that NFS will be needed to crack this number.

[QUOTE=Mr. P-1;333784]No. P-1 will find a factor 2kp+1 in stage 2 if there is a single prime in the factorisation of k between B1 and B2, and all other primes [I]and prime powers[/I] [/QUOTE](those last three words, I keep forgetting!)[quote]of the factorisation of k are < B1.

<snip>

This means that M1277 has been "TFed" by P-1 to 2*1277*2e12 which is about 52 bits, less than the real amount of TF.[/quote]Hmmm... could someone add to the [URL]http://www.mersenne.ca/exponent.php[/URL] so that it also shows the equivalent TF bit level that corresponds to the highest P-1 B1 bound done so far?

(Or maybe I should get energetic and learn PHP so I could add that myself ... Okay, it's on my to-do list.)

[QUOTE=Mr. P-1;333784]No. P-1 will find a factor 2kp+1 in stage 2 if there is a single prime in the factorisation of k between B1 and B2, and all other primes [I]and prime powers[/I] of the factorisation of k are < B1. Theoretically k could be a prime power a little over 2*10^12 and the factor will have been missed. This means that M1277 has been "TFed" by P-1 to 2*1277*2e12 which is about 52 bits, less than the real amount of TF.

*snipped*[/QUOTE]

Would (1e13+37) have been found, then? By "Theoretically k could be a prime power a little over 2*10^12 and the factor will have been missed", do you mean, say, k=2^41 would have been missed? Thanks for clarification!

P-1 is a factoring method based on Fermat's theorem. To be more clear, when we apply this to mersenne numbers, we should call it "q-1" factoring method, as we normally use p to denote the exponent, and it is confusing, specially for newcomers.

So, Fermat says that for a prime q, and a base b which is not multiple of q, we always have b[SUP]q-1[/SUP]=1 (mod q).

This implies the fact that b[SUP]s*(q-1)[/SUP]=1[SUP]s[/SUP]=1 (mod q) for any natural s. Or, we can say that b[SUP]s*(q-1)[/SUP]-1 is a multiple of q, note this number z=q*x.

Now assume q is a factor of some mersenne number Mp=2[SUP]p[/SUP]-1.

This means Mp=q*y. So, if we do a GCD of Mp and z=b[SUP]s*(q-1)[/SUP]-1=qx, we find the common factor q.

How can we find z? Generally, we can not find it. What "q-1" algorithm is doing, we take a random b (for P95 this is 3) and we raise it at some power a lot of times, computing y=(((...(((b^p[SUB]1[/SUB])^p[SUB]2[/SUB])^p[SUB]3[/SUB])...)...)...)^p[SUB]n[/SUB] and we [B]might be lucky[/B] that the product of all p[SUB]i[/SUB] contains all factors of q-1, and other numbers (whose product is s), so we have our z=b[SUP]s*(q-1)[/SUP]. If so, the gcd of y-1(=z) and Mp will reveal the factor.

For this to happen, ALL factors of q-1 must be contained in the product of those p[SUB]i[/SUB].

For the case of mersenne numbers, we already know the fact that any factor q has the form q=2*k*p+1, for some natural k. So, q-1 is in fact 2*k*p. So, by already considering p[SUB]0[/SUB]=2 and p[SUB]1[/SUB]=p in the product above, we reduce the problem from "factors of q-1" to "factors of k".

So, rephrase: For this to happen, ALL factors of k must be contained in the product of those p[SUB]i[/SUB], for i=3 to n.

I said all factors, not all prime factors. This means prime factors, and all their powers, and all their combinations.

What "q-1" algorithm came with new, is an efficient way to compute b at a power E equal to LCM(1, 2, 3, .... B1). So, like taking all natural numbers up to B1, prime or not, compute their product in an efficient way, take that LCM of them, be this E, and use this number as "s*(q-1)" given above. Or s*k, if you like, as we already added 2 and p to the product. Of course, this number is a product of many-many-many "small" numbers, and IF k has nothing but small factors, whose powers are under B1, then k is contained in this number, and we really have E=s*k for some s. In this case b[SUP]E*k[/SUP]-1 is a multiple of q, and the GCD will reveal a factor.

This is what [B]Stage 1[/B] of "q-1" algorithm is doing. It will find a factor of Mp if all factors of k raised at their powers are small enough (under B1).

Statistically, most of the composite numbers (k in our case) have a single factor which is much larger comparing with the other factors (this is because only one factor, maximum, can be over sqrt(k), and there are more primes between sqrt(k) and k/2 comparing with primes between 2 and sqrt(k), etc, for example, if k=10000, there are 25 primes under 100, and 644 primes between 100 and 5000, which will give 644 numbers of the form 2*prime<10k, also there are 445 primes between 100 and 3333, which give 445 primes of the form 3*prime<10k, then 342 numbers which are 4*prime, then 278 which are 5*prime, etc.

So, most of the numbers have a big factor, and then nothing but small factors.

As we already have computed c=b[SUP]E[/SUP] and did not find a factor, we can try our luck by randomly picking a large prime, p[SUB]x[/SUB] between B1 and a larger limit B2, and compute c^p[SUB]x[/SUB]. If p[SUB]x[/SUB] is "that big factor of k", we are done. If not, we may pick another, and try again, compute c^p[SUB]y[/SUB], etc.

What [B]Stage 2[/B] of "q-1" is doing, it comes with an efficient way to raise c at all primes between B1 and B2 by doing cheap multiplications instead of expensive powering.

So, stage 2 will find a factor q=2kp+1 of Mp, when all factors of k raised ar their powers are smaller then B1, except a single factor which can be between B1 and B2.

(still some lunch break left, but reserved for corrections :smile:, expecting axn or some others to jump...:razz:)

[QUOTE=c10ck3r;333806]Would (1e13+37) have been found, then? By "Theoretically k could be a prime power a little over 2*10^12 and the factor will have been missed", do you mean, say, k=2^41 would have been missed? Thanks for clarification![/QUOTE]

Yes and yes.

[URL="http://www.mersenne.org/report_exponent/?exp_lo=61829237"]61829237[/URL] has a factor(P-1): [URL="http://www.mersenne.ca/exponent/61829237"]788336251820136770338211725470049[/URL]
109.28 bits. :banana:

P-1 found a factor in stage #2, B1=585000, B2=10530000.
M[URL="http://www.mersenne.ca/exponent/62407999"]62407999[/URL] has a factor: 3961718768062037610391289 (81.712 bits)

k=2^[SIZE=2]2 × 7 × 19 × 1583 × 7001 × 5383451[/SIZE]

This is my first factor in 60M range and my largest prime factor so far.