Euler's theorem

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, then is congruent to modulo n, where denotes Euler's totient function; that is

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 not prime.[2]

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 some of Carmichael's theorems.

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.

  1. ^ See:
  2. ^ See:
    • L. Euler (published: 1763) "Theoremata arithmetica nova methodo demonstrata" (Proof of a new method in the theory of arithmetic), Novi Commentarii academiae scientiarum Petropolitanae, 8 : 74–104. Euler's theorem appears as "Theorema 11" on page 102. This paper was first presented to the Berlin Academy on June 8, 1758 and to the St. Petersburg Academy on October 15, 1759. In this paper, Euler's totient function, , is not named but referred to as "numerus partium ad N primarum" (the number of parts prime to N; that is, the number of natural numbers that are smaller than N and relatively prime to N).
    • For further details on this paper, see: The Euler Archive.
    • For a review of Euler's work over the years leading to Euler's theorem, see: Ed Sandifer (2005) "Euler's proof of Fermat's little theorem" Archived 2006-08-28 at the Wayback Machine