Densities of EHS numbers and Pillai primes: do the counting ratios converge, and to what? (Erdős #1074)
Statement
Let $S$ be the set of all integers $m\geq 1$ (the 'EHS numbers') for which there exists a prime $p\not\equiv 1\pmod{m}$ with $m!+1\equiv 0\pmod{p}$. Does the limiting density $$\lim_{x\to\infty}\frac{\lvert S\cap[1,x]\rvert}{x}$$ exist, and if so what is its value? Similarly, let $P$ be the set of primes $p$ (the 'Pillai primes') for which there exists an integer $m$ with $p\not\equiv 1\pmod{m}$ and $m!+1\equiv 0\pmod{p}$. Does $$\lim_{x\to\infty}\frac{\lvert P\cap[1,x]\rvert}{\pi(x)}$$ exist, and what is its value?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred, else full written) that the density of $S$, $\lim|S\cap[1,x]|/x$, exists together with its value (Hardy–Subbarao conjecture: $1$), and/or that the relative density of $P$ in the primes, $\lim|P\cap[1,x]|/\pi(x)$, exists together with its value; each of the two limits settled with proof is a fully-resolving result for that part. ADVANCES, each independently checkable: a proof that $S$ has density $1$ (or any proven lower bound on its lower density strictly above what is currently known); a proven bound on the relative density of $P$; or a reproducible exact computation of $S$ and $P$ up to a new height far beyond $2^{10}$, delivered with the program and the empirical density curves extending A064164 / A063980 (state $2^{10}$, the Hardy–Subbarao computation height, as the bar). Deliver the proof or the computation code plus the attained height and the tabulated densities.
Background
Questions of Erdős, Hardy and Subbarao [HaSu02] — who named $S$ the 'EHS numbers' and $P$ the 'Pillai primes' and proved both sets infinite — mentioned in problem A2 of Guy's Unsolved Problems in Number Theory; listed as open on erdosproblems.com/1074 (fetched 2026-07-21, status 'open'). Pillai [Pi30] first asked whether $P$ is non-empty; Chowla answered yes, noting e.g. $14!+1\equiv 18!+1\equiv 0\pmod{23}$, so $23\in P$. The sequences begin $S:8,9,13,14,15,16,17,\ldots$ (OEIS A064164) and $P:23,29,59,61,67,71,\ldots$ (OEIS A063980). Hardy and Subbarao computed all EHS numbers up to $2^{10}$; from the data — EHS numbers clustering in long runs of consecutive integers — they conjectured the density of $S$ exists and equals $1$, a view Erdős came to share. For $P$ the observed relative density lay between $0.5$ and $0.6$, though they suspected it might tend to $1$ very slowly and non-monotonically. Neither limit is proven to exist. A Lean-formalised statement is available in the formal-conjectures project; the problem is companion to Erdős #1072 and #1073 (erdosproblems.com/1072, erdosproblems.com/1073). Attacker's tool: extend the exact computation of $S$ and $P$ far beyond $2^{10}$ by testing, for each $m$ (resp. $p$), the factorial congruence $m!\equiv-1\pmod p$ jointly with $p\not\equiv1\pmod m$, then track the empirical densities and their trends to inform or refute the conjectured limits; complement with sieve/analytic arguments for the density of $S$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1074 (T. F. Bloom) | website |
| REF-02 | OEIS A064164 — EHS numbers (the set S) | website |
| REF-03 | OEIS A063980 — Pillai primes (the set P) | website |
| REF-04 | Lean formalisation of Erdős #1074 (formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.