In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and is Euler's totient function, then a raised to the power is congruent to 1 modulo n; that is 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. 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 not prime. The converse of Euler's theorem is also true: if the above congruence is true, then and must be coprime. The theorem is further generalized by Carmichael's theorem. The theorem may be used to easily reduce large powers modulo . For example, consider finding the ones place decimal digit of , i.e. . The integers 7 and 10 are coprime, and . So Euler's theorem yields , and we get . In general, when reducing a power of modulo (where and are coprime), one needs to work modulo in the exponent of : if , then . 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.
Dimitri Stelio Wyss, Michael Gröchenig