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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 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.
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 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.
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;
The cototient of is defined as (), i.e. the number of positive integers less than or equal to that have at least one prime factor in common with .For example, the cototient of 6 is 4 since these four positive integers have a prime factor in common with 6: 2, 3, 4, 6.
I think it would be better is a full proof for the multiplicity of the totient function is furnished in this page. If someone disagrees, please say so. I will put up the proof 3 days after this message is posted if there is no opposition — Preceding unsigned comment added by NKRVVI (talk • contribs) 10:49, 16 September 2021 (UTC)