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Claim. A. Flammenkamp and F. Luca, Infinite families of noncototients, Colloq. Math. 86 (2000), 37–41, received 2 July 1999, the date this page carries. A noncototient is a positive integer $n$ for which $x-\phi(x)=n$ has no solution. The paper's Proposition: if $k$ is an odd prime that is not a Mersenne prime, $2^tk-1$ is composite for every $t\ge1$ (a Riesel number), and $2k$ is a noncototient, then $2^mk$ is a noncototient for every $m\ge1$. The paper's Theorem: for every $m\ge1$, each $2^mk$ with $k\in\{509203, 2554843, 9203917, 9545351, 10645867, 11942443, 65484763\}$ is a noncototient. The Riesel condition comes from covering systems of congruences, and the condition on $2k$ is checked by computer: a solution $x$ of $x-\phi(x)=2k$ is even and squarefree and lies in $[2.39k,4k]$ in the tested range. Either family answers yes to Problem 418: infinitely many positive integers are not of the form $n-\phi(n)$. The value $k=509203$ is Browkin and Schinzel's, so the theorem extends their result (claim page) with six further values.

Depends on. No page of this wiki.

Acceptance. Refereed: Colloq. Math. 86 (2000), 37–41. The site's discussion thread links the paper (comment of 21 November 2025), but the site's commentary does not credit it, so no reviewed evidence is listed.