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.
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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 The complexity of the original algorithm of Agrawal et al. Bailey, D. Experimental Mathematics in Action.
Bernstein, D. Bernstein D. Berrizbeitia, P. Borwein, J. Experimentation in Mathematics: Computational Paths to Discovery. Bornemann, F. Clark, E. Crandall, R. Prime Numbers: A Computational Perspective, 2nd ed.
New York: Springer-Verlag, Germundsson, R. Granville, A. Indian Institute of Technology. Kanpur, India: Indian Institute of Technology, Lenstra H.
March Pomerance, C. Robinson, S. A16, August 8, Wagon, S. Mathematica in Action. New York: W. Freeman, pp. Weisstein, E.
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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 AKS primality test
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. 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.