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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. [40] A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient.
where ζ(2) is the Riemann zeta function for the value 2, which is [1] ¶. Φ( n ) is the number of coprime integer pairs {p, q}, 1 ≤ p ≤ q ≤ n . The summatory of reciprocal totient function
A highly totient number is an integer that has more solutions to the equation () =, where is Euler's totient function, than any integer smaller than it. The first few highly totient numbers are The first few highly totient numbers are
Euler's theorem Euler's theorem states that if n and a are coprime positive integers, then a φ(n) is congruent to 1 mod n. Euler's theorem generalizes Fermat's little theorem. Euler's totient function For a positive integer n, Euler's totient function of n, denoted φ(n), is the number of integers coprime to n between 1 and n inclusive.
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 values that a cyclotomic polynomial () may take for other integer values of x is strongly related with the multiplicative order modulo a prime number. More precisely, given a prime number p and an integer b coprime with p , the multiplicative order of b modulo p , is the smallest positive integer n such that p is a divisor of b n − 1 ...
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 ...
The exponent of the group, that is, the least common multiple of the orders in the cyclic groups, is given by the Carmichael function (sequence A002322 in the OEIS). In other words, λ ( n ) {\displaystyle \lambda (n)} is the smallest number such that for each a coprime to n , a λ ( n ) ≡ 1 ( mod n ) {\displaystyle a^{\lambda (n)}\equiv 1 ...