YAFA (c) JGW 2008

YAFA=Yet another factorization algorithm :)
I don't know whether this is any good (I _think_ it's new) - comments welcome...
Suppose we wish to factor N.
Write N=x^(2^n)-1, s.t. x=(N+1)^(2^-n)
Then N=(x-1)(x+1)(x^2+1)(x^4+1)...(x^2^(n-1)+1)
Start w/ n=1, and evaluate all the 2^n combinations of products of brackets above, testing for exact division into N as you go.
After all unsuccessful 2^n attempts, increment n, and repeat.

(M+2)1257787

Mersenneplustwo project news:
Following up the recent find of the 8-digit factor of (M+2)1257787 by random-based WEP (factorize3), I also confirmed this result with "sequential" WEP (factorize2).
The factor is found on the first attempted base after the factor of 3 comes out:
"
[james@localhost wec]$ time ./factorize2.gmp4.2.1 -Pp1257787 -a0
a= 5031149 factor=3
a= 7546723 factor=20124593


real 16844m25.553s
user 15836m58.317s
sys 363m38.512s

"

FermatSearch

The FERMATSEARCH.ORG (searching for factors of Fermat numbers) project has a new and improved website!
http://www.fermatsearch.org/index.html
Its most recently found factor (August 21, 2007) was:
"485.2^338297+1 divides F338295"
Note that for really huge Fermats like this, the only efficient method of searching for factors is modular trial-division. However there is also some ECM work going on on the smaller Fermats... It will be interesting to see cooperation with the new Primenet (factorization) server?

PrimeNet v5

Here's an exciting bit of news:
It looks like v5 of the GIMPS server will have built-in automatic reporting of factorization effort on Mersenne (and Fermat) numbers. As with the prime search itself, George Woltman and Scott Kurowski seem to have done a great job!
http://v5www.mersenne.org/report_ECM/

Bernoulli and Euler Numbers

Some more recent news culled from the mersenneforum :)
http://mersenneforum.org/showthread.php?t=9841
advertising the recent revival of a couple of (related) factorization projects I had previously overlooked:
Sam Wagstaff (of Cunningham project fame) leads an initiative to find factors of Bernoulli, and Euler composites at:
http://homes.cerias.purdue.edu/~ssw/bernoulli/index.html
Note that Bruce Dodson has already found some good-sized ECM factors from these targets, presumably as a diversion from the main Cunningham project!
Some links describing these numbers:
http://en.wikipedia.org/wiki/Bernoulli_number
http://en.wikipedia.org/wiki/Euler_number

GMP-EECM

Bernstein, Birkner, Lange and Peters present GMP-EECM, a variant of GMP-ECM (and developed from it) but using Edwards curves, rather than the more standard Montgomery curves of regular GMP-ECM.
http://cr.yp.to/papers.html#eecm
In one of their tests, this reduced the stage 1 time from 2448mS to 2276mS.
Alex Kruppa has stated that this improvement will also be incorporated into some future version of GMP-ECM.

Msieve v1.33

Msieve v1.33 now available:
Announcement at:
http://mersenneforum.org/showpost.php?p=122776&postcount=402
Download from:
http://www.boo.net/~jasonp/qs.html

Harpertown

Intel officially announces availability of its new "Penryn" chips.
http://www.bit-tech.net/news/2008/01/07/intel_announces_16_new_45nm_cpus_at_ces/1
Apple has already incorporated several server versions, including quad-cores, into the Mac Pro line.

Eric Landquist

[picture from Wikipedia - copyright Just H, creative commons license]

Here is a link to Eric Landquist's paper on the NFS:
http://www.math.uiuc.edu/~landquis/nfsieve.pdf
Quote from the above:
"At least one author [17] believes that there
are faster methods out there. It’s my life-long dream to find one. I also
hope to see the Red Sox win a World Series."
Well, the Red Sox have won _two_ World Series since the paper was published (2002) - though personally I am an Astros fan! :)

Frequency of Posts 2

Having made it to the New Year ( :) ), I expect the frequency of my posts to drop off somewhat, as I'm now running out of useful things to say...

M2008

To celebrate New Year 2008, for the past few days I have been factorizing M2008.
Here are the results:
ibu:~/math james$ time java superfac9
f2^2008-1

[29392145799020915820360529950148658790971333173470597132227654062739616291644680034730482849702560509912216694758079047000246245398094216484503842717866321546017277221199943680176327461949451487085805309456252478664093558693475421170513158666359386616551679118889574095089825179039567782281258040824405166424107240700021377434209148110825999078639302784109824695476896212613634081852488010690884578129204889342821483040517575643751434792922414912394467695078935531662069192598956042024980981047457429185377388949433859975257289323374605954282310600673952044911495373010647749329399156163119321894151520255]
wanless...
brutep: 3
wanless...
brutep: 5
wanless...
brutep: 17
brute...
brutep: 503
wanless...
ecm...
aprtcle: 54217
ecm...
aprtcle: 238451
ecm...
aprtcle: 178230287214063289511
ecm...
aprtcle: 61676882198695257501367
ecm...
aprtcle: 12070396178249893039969681
aprtcle:
5058345723951854688505665428846313806490903121677364358901199128608233
wanless...
brute...
aprtcle: 5021
ecm...
ecm...
aprtcle: 1972386557777
aprtcle: 1912621
ecm...
aprtcle: 57762875981
ecm...
aprtcle: 38508212572597
ecm...
aprtcle: 45063180240128066017730357
siqs...
aprtcle: 15992518154179475674328213556857438690614816129
aprtcle: 86245368961389419078481015822433
aprtcle:
10084786891164868903044000461741193511166162933699139834764709537603304010587634094053631800618313958847949862753441381884114308571221779233867837717363364521350181435128687986278658451823408133532192171438161388219841827075198840055685217831601941566655243743704980863680404547437764128787958275830001
duration: 40323 seconds