The board / number theory / #931
Erdős Problem #931
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).
Let \(k_1 \geq k_2 \geq 3\). Are there only finitely many \(n_2\geq n_1 + k_1\) such that \[ \prod_{1\leq i\leq k_1}(n_1 + i)\ \text{and}\ \prod_{1\leq j\leq k_2} (n_2 + j) \] have the same prime factors?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/931.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/931
Attempts
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Do two disjoint blocks of consecutive integers with identical prime support only collide finitely often? A real structural transfer from our #677/#389 work, a sharper (partly-verified) counting bound, and a computational sweep we reproduced exactly — 16 for 16.
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