# Teorema di Eulero In number theory, Teorema di Eulero (also known as the Fermat–Euler theorem or Euler's totient theorem) afferma che, if n and a are coprime positive integers, e {stile di visualizzazione varphi (n)} is Euler's totient function, then a raised to the power {stile di visualizzazione varphi (n)} è congruente a 1 modulo n; questo è {stile di visualizzazione a^{varfi (n)}equivalente 1{pmod {n}}.} In 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. Successivamente, 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, poi {stile di visualizzazione a} e {stile di visualizzazione n} must be coprime.

The theorem is further generalized by Carmichael's theorem.

The theorem may be used to easily reduce large powers modulo {stile di visualizzazione n} . Per esempio, consider finding the ones place decimal digit of {displaystyle 7^{222}} , cioè. {displaystyle 7^{222}{pmod {10}}} . The integers 7 e 10 are coprime, e {stile di visualizzazione varphi (10)=4} . So Euler's theorem yields {displaystyle 7^{4}equivalente 1{pmod {10}}} , and we get {displaystyle 7^{222}equiv 7^{4volte 55+2}equivalente (7^{4})^{55}times 7^{2}equiv 1^{55}times 7^{2}equiv 49equiv 9{pmod {10}}} .

In generale, when reducing a power of {stile di visualizzazione a} modulo {stile di visualizzazione n} (dove {stile di visualizzazione a} e {stile di visualizzazione n} are coprime), one needs to work modulo {stile di visualizzazione varphi (n)} in the exponent of {stile di visualizzazione a} : Se {displaystyle xequiv y{pmod {varfi (n)}}} , poi {stile di visualizzazione 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.