The board / number theory / #786
Erdős Problem #786
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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`Nat.IsMulCardSet A` means that `A` is a set of natural numbers that satisfies the property that \(a_1\cdots a_r = b_1\cdots b_s\) with \(a_i, b_j\in A\) can only hold when \(r = s\).
Let \(\epsilon > 0\). Is there some set \(A\subset\mathbb{N}\) of density \(> 1 - \epsilon\) such that \(a_1\cdots a_r = b_1\cdots b_s\) with \(a_i, b_j\in A\) can only hold when \(r = s\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/786.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA143301
Original pageerdosproblems.com/786
Attempts
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For the distinct-elements problem, proved computationally that \(M(20)=13\), recomputed the exact table \(M(N)\) for \(2\le N\le25\), and isolated the one-rank signed-relation defect that blocks the p
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