# noncototient

An integer $n>0$ is called a if there is no solution to $x-{\phi}(x)=n$, where ${\phi}(x)$ 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 $pq-{\phi}(pq)=pq-(p-1)(q-1)=p+q-1$, an odd number if $2.

Title noncototient Noncototient 2013-03-22 15:55:48 2013-03-22 15:55:48 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A25 Nontotient