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
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where
The converse of Euler's theorem is also true: if the above congruence is true, then a and n must be coprime.
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 ones place 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 ≡ 74 × 55 + 2 ≡ (74)55 × 72 ≡ 155 × 72 ≡ 49 ≡ 9 (mod 10).
Note that if
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 ax ≡ ay (mod n).Euler's theorem also forms the basis of the RSA encryption system, where the net result of first encrypting a plaintext message, then later decrypting it, amounts to exponentiating a large input number by kφ(n) + 1, for some positive integer k. Euler's theorem then guarantees that the decrypted output number is equal to the original input number, giving back the plaintext.
Proofs
1. Euler's theorem can be proven using concepts from the theory of groups: The residue classes (mod n) that are coprime to n form a group under multiplication (see the article Multiplicative group of integers modulo n for details.) The order of that group is φ(n) where
2. There is also a direct proof: Let R = {x1, x2, ..., xφ(n)} be a reduced residue system (mod n) and let a be any integer coprime to n. The proof hinges on the fundamental fact that multiplication by a permutes the xi: in other words if axj ≡ axk (mod n) then j = k. (This law of cancellation is proved in the article multiplicative group of integers modulo n.) That is, the sets R and aR = {ax1, ax2, ..., axφ(n)}, considered as sets of congruence classes (mod n), are identical (as sets - they may be listed in different orders), so the product of all the numbers in R is congruent (mod n) to the product of all the numbers in aR:
Euler quotient
The Euler quotient of an integer a with respect to n is defined as:
The special case of Euler quotient is Fermat quotient, it happens when n is prime.
A number n coprime to a which divides
The least base b > 1 which n is a Wieferich number are
2, 5, 8, 7, 7, 17, 18, 15, 26, 7, 3, 17, 19, 19, 26, 31, 38, 53, 28, 7, 19, 3, 28, 17, 57, 19, 80, 19, 14, 107, 115, 63, 118, 65, 18, 53, 18, 69, 19, 7, 51, 19, 19, 3, 26, 63, 53, 17, 18, 57, ... (sequence A250206 in the OEIS)