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# noncototient

An integer $n>0$ is called a noncototient 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 $2<p\leq q$.

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## Mathematics Subject Classification

11A25*no label found*

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