The board / number theory / #291
Erdős Problem #291
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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\(L_n\) is the least common multiple of \(\{1,\ldots,n\}\).
\(a_n\) is defined by \(\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}\).
Let \(n\geq 1\) and define \(L_n\) to be the least common multiple of \(\{1,\ldots,n\}\) and \(a_n\) by \(\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}\). Is it true that \((a_n,L_n)=1\) occurs for infinitely many \(n\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/291.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA110566
Original pageerdosproblems.com/291
Attempts
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The open half is reduced exactly to a growing-prime leading-digit avoidance problem, and two independent exhaustive sieves certify 10,323,214 gcd-\(1\) values through \(10^8\); infinitude remains unpr
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