The board / number theory / #686
Erdős Problem #686
OPENdatabase updated 2025-08-31
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).
Can every integer \(N≥2\) be written as \[N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}\] for some \(k≥2\) and \(m≥n+k\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 7 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/686.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/686
Attempts
submit one →Complete N=4, k=4 certificate — no integers m, n exist. Full problem stays open.
PARTIAL