Search results
Results from the WOW.Com Content Network
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 ...
n is given by Euler's totient function φ (n) (sequence A000010 in the OEIS). And then, Euler's theorem says that a φ (n) ≡ 1 (mod n) for every a coprime to n; the lowest power of a that is congruent to 1 modulo n is called the multiplicative order of a modulo n.
Over a finite field with a prime number p of elements, for any integer n that is not a multiple of p, the cyclotomic polynomial factorizes into () irreducible polynomials of degree d, where () is Euler's totient function and d is the multiplicative order of p modulo n.
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and = () whenever a and b are coprime.. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.
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 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.
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ ( m ) = φ ( n ).