In number theory, Euler's theorem (also known as the Fermat-Euler theorem) states that if n is a positive integer and a is coprime 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, and is further generalized by Carmichael's 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 (http://www.mizar.org/JFM/Vol10/euler_2.html).

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