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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 ...
Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 … p k where the p i are distinct primes, then φ(n) = (p 1 − 1)(p 2 − 1)...(p k – 1).
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.
The degree of , or in other words the number of nth primitive roots of unity, is (), where is Euler's totient function. The fact that Φ n {\displaystyle \Phi _{n}} is an irreducible polynomial of degree φ ( n ) {\displaystyle \varphi (n)} in the ring Z [ x ] {\displaystyle \mathbb {Z} [x]} is a nontrivial result due to Gauss . [ 4 ]
An average order of σ(n), the sum of divisors of n, is nπ 2 / 6; 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;
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by ():= = (),.It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n.
In mathematics, specifically number theory, a sparsely totient number is a natural number, n, such that for all m > n, > ()where is Euler's totient function.The first few sparsely totient numbers are: