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A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which φ(n) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation. [41] A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient.
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 ...
In fact, the Disquisitiones contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. A primitive root exists if and only if n is 1, 2, 4, p k or 2p k, where p is an odd prime and k > 0. For all other values of n the multiplicative group of integers modulo n is not cyclic.
The definition of the multiplicative order implies that, if n is the multiplicative order of b modulo p, then p is a divisor of (). The converse is not true, but one has the following. If n > 0 is a positive integer and b > 1 is an integer, then (see below for a proof) =, where
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are: Some other reduced residue systems modulo 12 are: {13,17,19,23}
The number of primitive elements in a finite field GF(q) is φ(q − 1), where φ is Euler's totient function, which counts the number of elements less than or equal to m that are coprime to m. This can be proved by using the theorem that the multiplicative group of a finite field GF( q ) is cyclic of order q − 1 , and the fact that a finite ...
Euler's theorem: If a and m are coprime, then a φ(m) ≡ 1 (mod m), where φ is Euler's totient function. A simple consequence of Fermat's little theorem is that if p is prime, then a −1 ≡ a p−2 (mod p) is the multiplicative inverse of 0 < a < p. More generally, from Euler's theorem, if a and m are coprime, then a −1 ≡ a φ(m)−1 ...