Msieve V1.31

New version of msieve (by Jason Papadopoulos) now available. It seems this is a significant upgrade, with lots of new stuff.
Download from:
http://www.boo.net/~jasonp/qs.html
Announcement here:
http://mersenneforum.org/showpost.php?p=120642&postcount=339

CUDA

Here are some links to an interesting new technology, which might be very useful for any "embarassingly parallel" (see earlier blog-post for definition of that) factorization (or other) algorithms. It's basically a method of using all the processing power inherent in modern GPUs for CPU-type calculations, and something I for one will be keeping an eye on...
http://en.wikipedia.org/wiki/CUDA
http://developer.nvidia.com/object/cuda.html#downloads
http://courses.ece.uiuc.edu/ece498/al1/
http://courses.ece.uiuc.edu/ece498/al1/Syllabus.html
It's even coming to Mac! :)
http://forums.nvidia.com/index.php?showtopic=47884

GMP-M+2

I've recently noticed an interesting feature of GMP's implementation of mod_pow (ie modular exponentiation). I'm using this as part of random-base WEP on large M+2's. Anyway I always imagined, and hoped it would use "Russian Peasant" (see earlier post on this blog for description of that). If that is the case,
http://gmplib.org/manual/Powering-Algorithms.html#Powering-Algorithms
there should theoretically be a speed advantage to testing numbers that are (very) close to a power of 2 (as M+2's obviously are). And indeed I am observing just that - a 40% (!) speed increase testing the _raw_ M+2, rather than dividing out any factors (even though dividing out factors obviously leaves a slightly smaller number).

C138_101_75

I'm a total xyyxf junkie! :)
Recently started another GNFS factorization for them, this time of 138-digit input number.
ETA several months... [on 1 CPU](assuming all goes well). Currently sieved 9502/839184 rels. required.
Hopefully the process will further increase my understanding of GNFS, which is kindof the point!

WIFC

This entry also is probably somewhat overdue:
Check out Hisanori Mishima's comprehensive page on factorization at:
http://www.asahi-net.or.jp/%7EKC2H-MSM/mathland/matha1/
Note that there are also download links to factoring software (by Satoshi Tomabechi) on this page (especially MPQS-type factoring)

Cunningham Project

This is probably somewhat overdue, but here are a couple of links to the Cunningham Project, "one of the oldest continuously ongoing activities in computational number theory". [apparently it's been around since 1925]
http://en.wikipedia.org/wiki/Cunningham_project
http://homes.cerias.purdue.edu/~ssw/cun/index.html

So just how big are these numbers?

So just how big are these numbers?
Well mathematics is unique among the sciences for generating really SILLY (or impressive, depending on your point-of-view :) sized numbers. Even (standard) physics not only doesn't come close, but can't even stand comparison. Hence the other-worldly nature of math (and mathematicians! :)). Now possibly combinatorial aspects of holistic physics theories can generate big numbers, but in this case the theory is as much mathematical as physical, anyway.
The way math generates these numbers is by a trick called 'exponentiation' (usually). This allows one to write (and sometimes even calculate with and manipulate) a very long number in a very succinct form. Exponentiation is basically a shorthand notation for multiplying a number by itself many times. Thus 10^100 (also known as a googol) is the number you get by multiplying 10 by itself 100 times. If written out in full it is '1' followed by 100 zeroes. And this isn't even that big by math (or even factorization) standards. Some of the larger M+2 numbers I've been testing would have millions of digits if written out in full. You can see why I called these numbers 'silly'-sized! And if you want to go really crazy in math, you can even iterate/stack the exponents... (there are notations existent for this)
I expect many of you will be aware of the story of the Chinese inventor of chess? Apparently the emperor was so impressed with chess that he promised the inventor a reward based on the 64 squares of the chess-board, that the inventor (obviously a mathematician too) had tricked him into granting: namely this:
1 grain of rice for the first square, 2 for the second, 4 for the third, 8 for the fourth, and so on, doubling each time. Total number of grains of rice 2^64, or about 10^20 - far more than the whole production of China, and the emperor was never able to make good his promise...! This doubling process is another example of exponentiation, as the emperor clearly learned the hard way.
For comparison:
1) Number of seconds elapsed since Genesis ~ 2*10^11 (just a measly 12-digit number)
2) Number of seconds elapsed since the Jurassic ~ 2*10^15 (just a measly 16-digit number)
3) Number of seconds elapsed since the start of the visible universe ~ 3*10^20 (just a measly 21-digit number)
Incidentally, the complete factorization of any of these (exact) numbers on a modern PC would only take a (very) split-second.
I'll repeat...
Some of the numbers under test at the Mersenneplustwo project, for example, have _millions_ of digits. See why I used the adjective 'SILLY' yet - there really is almost no other word for these-size numbers?
Finally, and this is perhaps the most remarkable fact of all - mathematics also deals with the infinite. And compared to infinity (ie the _whole_ Universe) every single one of these numbers is actually infinitesimally SMALL! Now how do you get your head around that??? I know I for one, have problems with that...
Ahh, the mysteries of the finite and the infinite - or as Shakespeare famously put it:
"To be or not to be, that is the question"