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For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows: Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem. [3]
Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers.
Because of the sparsity of the Fermat numbers, the Pépin test has only been run eight times (on Fermat numbers whose primality statuses were not already known). [ 1 ] [ 2 ] [ 3 ] Mayer, Papadopoulos and Crandall speculate that in fact, because of the size of the still undetermined Fermat numbers, it will take considerable advances in ...
The false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2 340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2.
Chapter 5 generalizes Fermat's little theorem from numbers to polynomials, and introduces a randomized primality test based in this generalization. Chapter 6 provides the key mathematical results behind the correctness of the AKS primality test, and chapter 7 describes the test itself. [5]
Fermat primality test: A probabilistic test to determine whether a number is probable prime. 2001 Miller–Rabin primality test: Determines whether a given number is ...