Agrawal and colleagues announced a deterministic algorithm for determining if a number is prime that runs in polynomial time Agrawal et al. While this had long been believed possible Wagon , no one had previously been able to produce an explicit polynomial time deterministic algorithm although probabilistic algorithms were known that seem to run in polynomial time. Commenting on the impact of this discovery, P. Leyland noted, "One reason for the excitement within the mathematical community is not only does this algorithm settle a long-standing problem, it also does so in a brilliantly simple manner. Everyone is now wondering what else has been similarly overlooked" quoted by Crandall and Papadopoulos
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But problem with all of them is that they all are probabilistic in nature. So, here comes one another method i. Features of AKS primality test : 1. The AKS algorithm can be used to verify the primality of any general number given. The maximum running time of the algorithm can be expressed as a polynomial over the number of digits in the target number.
The algorithm is guaranteed to distinguish deterministically whether the target number is prime or composite. The correctness of AKS is not conditional on any subsidiary unproven hypothesis.
The AKS primality test is based upon the following theorem: An integer n greater than 2 is prime if and only if the polynomial congruence relation holds for some a coprime to n. Here x is just a formal symbol. The AKS test evaluates the equality by making complexity dependent on the size of r. This is expressed as which can be expressed in simpler term as for some polynomials f and g. This congruence can be checked in polynomial time when r is polynomial to the digits of n.
The AKS algorithm evaluates this congruence for a large set of a values, whose size is polynomial to the digits of n. The proof of validity of the AKS algorithm shows that one can find r and a set of a values with the above properties such that if the congruences hold then n is a power of a prime. As a should be co-prime to n. As the number increases, size increases. The code here is based on this condition and can check primes till
AKS Primality Test
It is not currently of any practical use. The former is good enough for almost all purposes, though there are different levels of testing people feel is adequate. This will be vastly faster than AKS and be just as correct in all cases. Almost all of the proof methods will start out or they should with a test like this because it is cheap and means we only do the hard work on numbers which are almost certainly prime. Moving on to proofs. In each case the resulting proof requires no conjectures, so these may be functionally compared.
Test de primalité AKS
Importance[ edit ] 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. 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. The maximum running time of the algorithm can be expressed as a polynomial over the number of digits in the target number.