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The Agrawal-Kayal-Saxena (AKS) primality test, discovered in 2002, is the first provably deterministic algorithm to determine the primality of a given number with a run time which is guaranteed to be polynomial in the number of digits, thus, given a large number , the algorithm will correctly determine whether that number is prime or not in time . (Many previous primality testing algorithms existed, but they were either probabilistic in nature, had a running time slower than polynomial, or the correctness could not be guaranteed without additional hypotheses such as GRH.)
The AKS test is of some relevance to the polymath project “Finding primes“, so I thought I would sketch the details of the test (and the proof that it works) here. (Of course, full details can be found in the original paper, which is nine pages in length and almost entirely elementary in nature.) It relies on polynomial identities that are true modulo when is prime, but cannot hold for non-prime as they would generate a large number of additional polynomial identities, eventually violating the factor theorem (which asserts that a polynomial identity of degree at most can be obeyed by at most values of the unknown).