Note - I threw this together in 2 minutes this morning, so it's far from complete!
Note - "Illegal" stuff here!

Sections

Prime hunting : | records | tetradics | drag racing | Phi(3,n) | AKS | puzzles |
Factoring & Sieving : | GFN sieving | sierpinski sieving | ECMNet | Factors near a googolplex

Prime Hunting

Phil's Big Head

At the time of writing, I'm discoverer or co-discoverer of the world's largest:

However, I've lost the crown on:

I'll flesh those out soon, providing the number, HREFs, etc.

Totally Tetradic! (NEW - 2004/04/26)

Project 101 - the hunt for tetradics.

Drag Racing Primes!

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!

Other Fields of Active Research

NEW!
I'm trying to create a definitive list of all primes of the form Phi(n,3), complete with full proofs

PRIMES in P

New-ish! (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 meta-FAQ, so to speak.

Some prime puzzles

Note - for prime puzzles, you want Carlos Rivera's Prime Puzzles Connection.

Prime Differences Triangle!

Consecutive primes with the same terminal digits

All Things Prime

Anyway, there is only one resource for prime-related info - Professor Caldwell's Prime Pages.

Non-primes

Taking Composites to Primes - i.e. factoring and sieving

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.

Sieving

Generalised Fermat Numbers - the next BIG thing

Ask Yves Gallot - he's spearheading the Generalised Fermat Search. I've volunteered to do some GFN pre-sieving 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 pre-sieving task ever undertaken! (There are larger sieving tasks, such as the ones to do with Brun's constant, and there have been some massive trial-division tasks (e.g. for GIMPS), but those aren't pre-sieving.)

Sierpinski Numbers

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.

Factorial Primes

I've done some factorial pre-sieving for Nuutti Kuosa's coordinated factorial prime search.

Factoring

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 hourly-updated mirror here.
Also check the ECMNet project's statistics page.

Factoring the seemingly unfactorable - numbers over a googolplex

here

Other Factoring Pursuits

I'm trying to put together a small, ultrafast library for factorisation of numbers 20 or so digits long using the "trivial" techniques (P-1, 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!

Wannabee Primes - PSPs and Carmichaels

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!

Euler's Phi, and Lehmer's Conjecture

Lehmer's conjecture says a number p with 

  phi(p) | (p-1)
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 

  (p-1) / phi(p)
such that the fraction is "small" in lowest terms.
I have some data here which I generated after seeing a fascinating discussion on sci.math between Jan Haugland and Erick Bryce Wong, which I rudely burst in on!

Other stuff

I've gathered together some things here:

Another hastily constructed page by Phil Carmody
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