Primes is in P

It appears to not be widely known, at least among some people who would like to know it, that primality testing can be done in polynomial time. The "naive" methods for determining whether a number is prime -- trial division and the like -- take time which is polynomial in the size of the input (the polynomial may be n^1/2, but it's still a polynomial) which is considered "large" because it's exponential in the number of digits or bits in the input.

The fact that primality testing can be done in polynomial time -- and without recourse to a randomized algorithm -- I've heard about here and there; not surprisingly, the exponent is large. Namely, to test n for primality takes time O~(log⁶ n), where O~(t(n)) = O(t(n) * P(log t(n))) and P is a polynomial. (If you don't like the O~ notation, you can say O(log^6+ε n) for any positive real ε.) It's conjectured that "6" can be replaced by "3". The proof is based on a generalization of Fermat's Little Theorem; of course it's nothing like the "naive" method of trial division, but the paper is surprisingly accessible given the strength of the result.

The paper is Manindra Agrawal, Neeraj Kayal, Nitin Saxena. PRIMES is in P. Annals of Mathematics, 160 (2004), 781-793. (http://www.math.princeton.edu/~annals/issues/2004/Sept2004/Agrawal.pdf)

Edited, 7:21 pm: It's been pointed out to me that this result is widely known among people who work in the appropriate fields, and that's true. But a lot of people who are just looking for some sort of color when they're giving an example of a hard algorithmic problem, and whose specialty is not in that area, either don't know this or don't remember it. I'm claiming that the result deserves to be more well-known among mathematicians in general.