# Théorème d'Euler In number theory, Théorème d'Euler (also known as the Fermat–Euler theorem or Euler's totient theorem) stipule que, if n and a are coprime positive integers, et {style d'affichage varphi (n)} is Euler's totient function, then a raised to the power {style d'affichage varphi (n)} est conforme à 1 modulo n; C'est {style d'affichage a^{varphi (n)}équiv 1{pmod {n}}.} Dans 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Ensuite, 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 not prime. The converse of Euler's theorem is also true: if the above congruence is true, alors {style d'affichage a} et {displaystyle n} must be coprime.

The theorem is further generalized by Carmichael's theorem.

The theorem may be used to easily reduce large powers modulo {displaystyle n} . Par exemple, consider finding the ones place decimal digit of {displaystyle 7^{222}} , c'est à dire. {displaystyle 7^{222}{pmod {10}}} . The integers 7 et 10 are coprime, et {style d'affichage varphi (10)=4} . So Euler's theorem yields {displaystyle 7^{4}équiv 1{pmod {10}}} , and we get {displaystyle 7^{222}equiv 7^{4fois 55+2}équiv (7^{4})^{55}times 7^{2}equiv 1^{55}times 7^{2}equiv 49equiv 9{pmod {10}}} .

En général, when reducing a power of {style d'affichage a} modulo {displaystyle n} (où {style d'affichage a} et {displaystyle n} are coprime), one needs to work modulo {style d'affichage varphi (n)} in the exponent of {style d'affichage a} : si {displaystyle xequiv y{pmod {varphi (n)}}} , alors {style d'affichage a^{X}equiv a^{y}{pmod {n}}} .

Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n being a product of two large prime numbers, and the security of the system is based on the difficulty of factoring such an integer.