The board / number theory / #363
Erdős Problem #363
DISPROVED LEANdatabase updated 2026-03-10
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A finite set of naturals is an interval of naturals.
A collection of intervals as in Erdős Problem 363: disjoint intervals of positive integers, each of size at least `4`, whose product is a square. Intervals containing `0` are excluded, since their product is `0`, which is a square.
Is it true that, for each fixed \(n\) and fixed sizes \(k_1,\ldots,k_n \geq 4\), there are only finitely many collections of disjoint intervals \(I_1,\ldots,I_n\) of size \(\lvert I_i\rvert = k_i\) for \(1\leq i\leq n\) such that\[\prod_{1\leq i\leq n}\prod_{m\in I_i}m\]is a square? The number of intervals and their sizes are fixed: the list `ks` of sizes determines both. Without this restriction finiteness fails, by a result of Skałba (see [Ul05]). This is false: Ulas [Ul05] constructed infinitely many such collections with \(n = 4\) and \(k_1 = \cdots = k_4 = 4\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/363.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/363
Attempts
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