Importance
AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional. Previous algorithms had been developed for centuries but achieved three of these properties at most, but not all four.
- The AKS algorithm can be used to verify the primality of any general number given. Many fast primality tests are known that work only for numbers with certain properties. For example, the Lucas–Lehmer test for Mersenne numbers works only for Mersenne numbers, while Pépin's test can be applied to Fermat numbers only.
- The maximum running time of the algorithm can be expressed as a polynomial over the number of digits in the target number. ECPP and APR conclusively prove or disprove that a given number is prime, but are not known to have polynomial time bounds for all inputs.
- The algorithm is guaranteed to distinguish deterministically whether the target number is prime or composite. Randomized tests, such as Miller–Rabin and Baillie–PSW, can test any given number for primality in polynomial time, but are known to produce only a probabilistic result.
- The correctness of AKS is not conditional on any subsidiary unproven hypothesis. In contrast, the Miller test is fully deterministic and runs in polynomial time over all inputs, but its correctness depends on the truth of the yet-unproven generalized Riemann hypothesis.
Read more about this topic: AKS Primality Test
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