The current world record for the largest known prime number was discovered on May 15, 2004, as part of the Great Internet Mersenne Prime Search (GIMPS). It has 7,235,733 digits. The previous record has 6,320,430 digits.
In 2002, the first algorithm was found that determines if a number is prime, and which is known to run in polynomial time - in this case, the length of time required is on the order of the number of digits of the number being tested, to the twelfth (12th) power.
The Online Encyclopedia of Integer Sequences will allow you to type in the first few terms of a sequence you suspect has a pattern, and will match it against any known integer sequence in its database. Several published results have a search of this database as an important initial component!
Numbers of the form 2(2n)+1 which are prime are called Fermat primes; they appear to be very rare. More generally, numbers of the form a(2n)+1 are called (surprise) generalized Fermat primes. There are some lists of generalized Fermat primes for (relatively) small values of a and n.
This page will compute the continued fraction expansion of any quadratic irrational of reasonable size (e.g, the integer in the square root should be less than 10,000,000,000).
This page lists all of the prine numbers up to 10 million.
A site called SOS Math at the Univ. of
Texas at El Paso offers pages of material on topics ranging from polynomial long
division, the quadratic formula, and trigonometric identities, to Taylor polynomials,
the Cauchy-Riemann equations, and Matrix algebra.
Another site covering similar material, including solved homework problems for you to practice on, is kept in Belgium.
Dan Sloughter has a web page containing Java programs for visualizing various mathematical concepts. My favorite is one which will draw the Taylor polynomial approximations for y=sin(x) .
A site called Karl's Calculus Tutor currently covers most of what would qualify as first-semester calculus, and some of the second semester, as well.
Forget a geometry formula? Check this page at Ask Dr. Math.
A Java-enabled page for generating Pascal's triangle (or rather, the last two digits of each entry, which is good enough through the 24th line). What's the pattern of the even numbers in the triangle?!