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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.
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 summatory of reciprocal totient function is defined as ():= = ()Edmund Landau showed in 1900 that this function has the asymptotic behavior (+ ) + + ()where γ is the Euler–Mascheroni constant,
That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even.
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 Carmichael function is named after the American mathematician Robert Carmichael who defined it in 1910. [1] It is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function. The order of the multiplicative group of integers modulo n is φ(n), where φ is Euler's totient function.
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 ...
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.