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For example, if a = 2 and p = 7, then 2 6 = 64, and 64 − 1 = 63 = 7 × 9 is a multiple of 7. 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.
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 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.
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 ...
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation + = if n is an integer greater than two (n > 2).. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. ... The special case where n is prime is known as Fermat's little theorem.
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to a n + b n = c n; Fermat's little theorem, a property of prime numbers; Fermat's theorem on sums of two squares, about primes expressible as a sum of ...
[2] [3] The statement that every prime p of the form + is the sum of two squares is sometimes called Girard's theorem. [4] For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640 ...