An integer n>0 is called a noncototientMathworldPlanetmath if there is no solution to x-ϕ(x)=n, where ϕ(x) is Euler’s totient function. The first few noncototients are 10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130 (listed in A005278 of Sloane’s OEIS).

Browkin and Schinzel proved in 1995 that there are infinitely many noncototients. What is still unknown is whether they are all even. Goldbach’s conjecture would seem to suggest that this is the case: given a semiprime pq, it follows that pq-ϕ(pq)=pq-(p-1)(q-1)=p+q-1, an odd numberMathworldPlanetmathPlanetmath if 2<pq.

Title noncototient
Canonical name Noncototient
Date of creation 2013-03-22 15:55:48
Last modified on 2013-03-22 15:55:48
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A25
Related topic NontotientMathworldPlanetmath