# 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[1] (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.[2] 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.

Gauss, Carl Friedrich; Clarke, Arthur A. (translated into English) (1986), Disquisitiones Arithemeticae (Deuxième, corrected edition), New York: Springer, ISBN 0-387-96254-9 Gauss, Carl Friedrich; Maser, H. (translated into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 Hardy, g. H; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Presse universitaire d'Oxford, ISBN 978-0-19-853171-5 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea External links Weisstein, Eric W. "Euler's Totient Theorem". MathWorld. Euler-Fermat Theorem at PlanetMath hide vte Leonhard Euler Euler–Lagrange equationEuler–Lotka equationEuler–Maclaurin formulaEuler–Maruyama methodEuler–Mascheroni constantEuler–Poisson–Darboux equationEuler–Rodrigues formulaEuler–Tricomi equationEuler's continued fraction formulaEuler's critical loadEuler's formulaEuler's four-square identityEuler's identityEuler's pump and turbine equationEuler's rotation theoremEuler's sum of powers conjectureEuler's theoremEuler equations (dynamique des fluides)Fonction d'EulerMéthode d'EulerNombre d'EulerNombre d'Euler (la physique)Euler–Bernoulli beam theoryNamesakes Category Portal: Mathematics Categories: Modular arithmeticTheorems in number theoryLeonhard Euler

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