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Statement

Problem G (p. 557). For a prime $p$, $f(p)$ is the least integer with $f(p)!+1\equiv0\pmod p$; it exists and is at most $p-1$ by Wilson's theorem. The authors write that they believe $f(p)=p-1$ for infinitely many $p$, but that the number of such $p\le x$ is probably $o(x/\log x)$. Below the problem the paper records that Erdős believed $f(p)/p\to0$ for almost all $p$.

Problem G is not starred, so by Section 3's preamble it was raised in discussion with Erdős.

Problem C on the same page is related: for $g(p)$, the number of $n<p$ with $n!+1\equiv0\pmod p$, it asks whether $\limsup g(p)=\infty$ and suggests $g(p)\to\infty$ for almost all $p$.

Proof pointer

An open problem; the paper proves nothing about it.

Read depth

Claims checked: Problems C and G and the sentence after G 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 1072: the problem's two questions, whether $f(p)=p-1$ for infinitely many $p$ and whether $f(p)/p\to0$ for almost all $p$, are the authors' belief in Problem G and Erdős's belief recorded after it. The paper states both as beliefs and proves nothing about them.