Note  I threw this together in 2 minutes this morning, so it's far
from complete!
Note  "Illegal" stuff here!
Prime hunting :
 records
 tetradics
 drag racing
 Phi(3,n)
 AKS
 puzzles

Factoring & Sieving :
 GFN sieving
 sierpinski sieving
 ECMNet
 Factors near a googolplex
At the time of writing, I'm discoverer or codiscoverer of the world's largest:
However, I've lost the crown on:
Project 101  the hunt for tetradics.
How many primes can you find on a sequence k*2^n+1 with a fixed k value? With some collaborators, I've found lots!. This section is in its infancy, and it probably changing from day to day! (And some of the amazing world records are changing from day to day too!). Do you feel the need, the need for speed? If so, come Drag Racing Primes with us!
NEW!
I'm trying to create a definitive list of all
primes of the form Phi(n,3), complete with full proofs
Newish! (it's at least current, anyway)
It's the latest news  Agrawal, Kayal, and Saxena have proved that unconditionally, primality can be determined in
polynomial time. Anyway, lots of questions are still being asked, so I've decided to gather together everything that
could be described as an answer and stick it all in one place. An AKS metaFAQ, so to speak.
Note  for prime puzzles, you want Carlos Rivera's Prime Puzzles Connection.
Consecutive primes with the same terminal digits
Anyway, there is only one resource for primerelated info  Professor Caldwell's Prime Pages.
Erm, not much here at the moment. However, a few sieving exploits are listed below. Most have little explanation  I'll flesh it out later... I intend to turn this into a bit of an explanation of how sieving works.
Ask Yves Gallot  he's spearheading the Generalised Fermat Search. I've volunteered to do some GFN presieving for the project, as it appeared in the past that everyone was doing sieving for their own little section  which is a waste.
NEW!  2003/01/13
Note  until someone provides evidence to the contrary, I believe that this
sieving task is the largest presieving task ever undertaken! (There are larger
sieving tasks, such as the ones to do with Brun's constant, and there have been
some massive trialdivision tasks (e.g. for GIMPS), but those aren't
presieving.)
After doing some initial sieving for the original Ballinger/Keller search, I've turned my hands to bigger things, and written an alternative sieve for the new Seventeen or Bust project which I'm hoping will reduce the workload for the whole task. Download binaries here.
I've done some factorial presieving for Nuutti Kuosa's coordinated factorial prime search.
The ECMNet client/server master server now provides some simple
statistics on how well it's progressing. For those that can't access
The server itself
there's an hourlyupdated mirror here.
Also check the
ECMNet project's
statistics page.
I'm trying to put together a small,
ultrafast library for factorisation of numbers 20 or so digits long using
the "trivial" techniques (P1, Rho, Squfof).
I intend to turn these into JavaScript and Java applets so that people can
experiment with the efficiency of the different algorithms.
I'm hoping to be the first person to force Shanks' Squfof onto the world!
(Yes, I know it's in LIP, but not enough people use LIP!)
SQUFOF SQUFOF SQUFOF  you heard it here first!
(second :) )
Oh, bugger, it's in Pari/GP too  third!
Carmichaels got me into number theory. On the Oxford University
mathematics entrance exam back in the early eighties, there was a
question about whether a^561 == a (mod 561). My teacher didn't know the
answer, and none of my group of mates knew the trick (Paul McGilly, Rashmi
Tank, Kedaar Kale  hey, get in touch, guys!) to prove it. However, while
visitting some event at London University, we pounced on the nearest Prof
and asked him to solve it for us. I believe it was Professor Sylvester
(no, not that Sylvester from London University!) who showed us in
about 3 seconds the trick. Amazing. It blew my mind. Since then I've got the
fuse replaced, but the joy of mathematics lives on!
I will put some actual content here eventually!
Lehmer's conjecture says a number p with
phi(p)  (p1)would be most interesting to find or disprove the existence of. He hints that it's probably impossible to do, which is nice!
A relaxed version of that is to find
(p1) / phi(p)such that the fraction is "small" in lowest terms.
I've gathered together some things here:
Another hastily constructed page by Phil Carmody
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