Problem 1074

Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$ . Does

\lim \frac{\lvert S\cap [1,x]\rvert}{x}

exist? What is it?

Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$ , then does

\lim \frac{\lvert P\cap [1,x]\rvert}{\pi(x)}

exist? What is it?

Source: erdosproblems.com/1074

Number theory

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Open

No claim settles this problem.

Gu04: Guy, Richard K., Unsolved problems in number theory, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. A2 "Primes connected with factorials", printed p. 12: Subbarao's Pillai primes, the primes $p$ with some $n$ having $n!+1\equiv0\pmod p$ and $p\not\equiv1\pmod n$ , the eight below $100$ , Hardy and Subbarao's infinitely many Pillai primes and EHS numbers, and the questions whether the Pillai primes have an asymptotic density (experimentally between $0.5$ and $0.6$ ) and what the density of the EHS numbers is; no proofs. Library home: Guy 2004.
HaSu02: Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554-559.
Pi30: S. S. Pillai, Question 1490. J. Indian Math. Soc. (1930), 230.

Problem 1074

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Proof claims

Discussion