The board / number theory / #1074
Erdős Problem #1074
OPENdatabase updated 2025-10-05
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The EHS numbers (after Erdős, Hardy, and Subbarao) are those \(m\geq 1\) such that there exists a prime \(p\not\equiv 1\pmod{m}\) such that \(m! + 1 \equiv 0\pmod{p}\).
The Pillai primes are those primes \(p\) such that there exists an \(m \ge 1\) with \(p\not\equiv 1\pmod{m}\) such that \(m! + 1 \equiv 0\pmod{p}\)
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 \[ \lim\frac{|S\cap[1, x]|}{x} \] exist?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 6 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1074.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1074
Attempts
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Exact EHS certificates give 120 members through 139 and isolate 140!+1; two independent scans give 21969 Pillai primes through 500000, but both density limits remain open.
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