noncototient

An integer $n>0$ is called a noncototient if there is no solution to $x-\varphi (x)=n$, where $\varphi (x)$ 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 $pq-\varphi (pq)=pq-(p-1)(q-1)=p+q-1$, an odd number if .

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	Nontotient