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.
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 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.
A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group of the field. Over GF(p) there are exactly φ(p m − 1) primitive elements and φ(p m − 1) / m primitive polynomials, each of degree m, where φ is Euler's totient function. [1]
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.
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 ...
In mathematics, a subset R of the integers is called a reduced residue system modulo n if: . gcd(r, n) = 1 for each r in R,R contains φ(n) elements,; no two elements of R are congruent modulo n.