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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 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 ]
Here φ denotes Euler's totient function. ... {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced ...
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;
Euler's totient or phi function, φ(n) is an arithmetic function that counts the number of positive integers less than or equal to n that are relatively prime to n. That is, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n which have no common factor with n other than 1.
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.