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Statement

Problem F* (p. 557). $A(x)$ is the number of composite $u<x$ for which $n!+1\equiv0\pmod u$; the print does not write the quantifier on $n$, and the examples fit the reading that some $n$ exists. The paper gives the examples $25$, $121$ and $721$ and asks (quoted): "Is $A(x)=o(x^\epsilon)$?" It does not write the quantifier on $\epsilon$.

Section 3 (p. 557) says the problems marked with an asterisk are original to Erdős; E* and F* carry it.

Read for every $\epsilon>0$, the question is whether $\log A(x)/\log x\to0$, that is $A(x)\le x^{o(1)}$.

Proof pointer

An open problem; the paper offers no bound or partial result. The examples check directly: $4!+1=25$, $5!+1=121=11^2$ and $6!+1=721=7\cdot103$.

Read depth

Claims checked: Problem F* and the preamble of Section 3 were read clause by clause on the page image of the print. Nothing here is independently reviewed.

Dependencies

None.

Source. G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109 (2002), no. 6, 554--559, doi:10.2307/2695445; the edition read is named on the source card.

Bears on

• Problem 1073: the problem asks whether $A(x)\le x^{o(1)}$ for the same count, with the existence of $n$ written out, which is Problem F* read for every $\epsilon>0$. The paper poses the question and proves nothing toward it.