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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 ...
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 ...
The following is a table of the Bell series of well-known arithmetic functions. The Möbius function has () =.; The Mobius function squared has () = +.; Euler's totient has () =.; The multiplicative identity of the Dirichlet convolution has () =
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
Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.
An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;