AKS primality test

The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P". The algorithm was the first to determine whether any given number is prime or composite within polynomial time. The authors received the 2006 Gödel Prize and the 2006 Fulkerson Prize for this work.

AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional. Previous algorithms had been developed for centuries and achieved three of these properties at most, but not all four.

While the algorithm is of immense theoretical importance, it is not used in practice. For 64-bit inputs, the Baillie–PSW primality test is deterministic and runs many orders of magnitude faster. For larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is far superior to AKS. Additionally, ECPP can output a primality certificate that allows independent and rapid verification of the results, which is not possible with the AKS algorithm.

The AKS primality test is based upon the following theorem: An integer n (≥ 2) is prime if and only if the polynomial congruence relation

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