In number theory, Euler's theorem (also known as the Fermat-Euler theorem) states that if n is a positive integer and a is relatively prime to n, then

a^φ(n) = 1 (mod n)

where φ(n) denotes Euler's totient function.

The theorem is a generalization of Fermat's little theorem.

The theorem may be used to easily reduce large powers modulo n. For example, consider finding the last decimal digit of 7²²², i.e. 7²²² mod 10. Note that 7 and 10 are coprime, and φ(10) = 4. So Euler's theorem yields 7⁴ = 1 (mod 10), and we get 7²²² = 7^{4·55 + 2} = (7⁴)⁵⁵·7² = 1⁵⁵·7² = 49 = 9 (mod 10).

In general, when reducing a power of a modulo n (where a and n are coprime), one needs to work modulo φ(n) in the exponent of a:

if x = y (mod φ(n)), then a^x = a^y (mod n).

Proofs of Euler's theorem

Leonhard Euler published a proof in 1736. Using modern terminology, one may prove the theorem as follows: the numbers a which are relatively prime to n form a group under multiplication mod n, the group of units of the ring Z/nZ. This group has φ(n) elements, and the statement of Euler's theorem follows then from Lagrange's theorem.

Another direct proof: if a is coprime to n, then multiplication by a permutes the residue classes mod n that are coprime to n; in other words (writing R for the set consisting of the φ(n) different such classes) the sets { x : x in R } and { ax : x in R } are equal; therefore, their products are equal. Hence, P = a^φ(n)P (mod n) where P is the first of those products. Since P is coprime to n, it follows that a^φ(n) = 1 (mod n).

The Mizar project has completely formalized and automatically checked a proof of Euler's theorem in the EULER_2 file.