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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]
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
Some of the proofs of Fermat's little theorem given below depend on two simplifications. The first is that we may assume that a is in the range 0 ≤ a ≤ p − 1 . This is a simple consequence of the laws of modular arithmetic ; we are simply saying that we may first reduce a modulo p .
Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a that is not a multiple of n; (typically, we choose a in the range 1 < a < n − 1). Calculate a n − 1 modulo n. If the result is not 1, then n is composite.
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.
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
Euler's Phi Function and the Chinese Remainder Theorem — proof that φ(n) is multiplicative Archived 2021-02-28 at the Wayback Machine; Euler's totient function calculator in JavaScript — up to 20 digits; Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions Archived 2021-01-16 at the Wayback Machine
by Fermat's little theorem, () (this property alone defines the weaker notion of probable prime to base a, on which the Fermat test is based); the only square roots of 1 modulo n are 1 and −1. Hence, by contraposition , if n is not a strong probable prime to base a , then n is definitely composite, and a is called a witness for the ...