The board / number theory / #873
Erdős Problem #873
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(a\) be some sequence of natural numbers. We set \(F(A,X,k)\) to be the count of the number of \(i\) such that \([a_i,a_{i+1}, \dots ,a_{i+k−1}] < X\), where the left-hand side is the least common multiple.
Let \(A = \{a_1 < a_2 < \dots\} \subseteq \mathbb{N}\) and let \(F(A,X,k)\) count the number of \(i\) such that \([a_i,a_{i+1}, \dots ,a_{i+k−1}] < X\), where the left-hand side is the least common multiple. Is it true that, for every \(\epsilon > 0\), there exists some \(k\) such that \(F(A,X,k) < X^\epsilon\)?
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/873.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/873
Attempts
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Proved the elementary uniform bound \(F(A,X,k)\ll_k X^{\Delta_k}\log X\) with explicit \(\Delta_k\downarrow1/4\), upgraded it to the literal live-page inequality for every \(\epsilon>1/4\), and exactl
PARTIAL