Problem 686

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claims/: The 2 claim pages of Problem 686, one per claimant's result; the problem's standing derives from them.

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Statement. Can every integer $N\geq 2$ be written as

$$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$

for some $k\geq 2$ and $m\geq n+k$?

Status. Open. The site's label is OPEN.

Source. erdosproblems.com/686, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #686, https://www.erdosproblems.com/686.

References.

• [Er79d] Erdős, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.

Formalization. Statement in formal-conjectures. A Lean proof of the non-square case is recorded on its claim page.

Current assessment

The question is open on the site. The non-square case is settled with $k=2$ by a Lean proof found by AlphaProof and merged into formal-conjectures, recorded on its claim page; this corpus has not built it. For squares, a note by Nat Sothanaphan, generated with GPT-5.4 Thinking and recorded on its claim page, claims that the squares representable with $k=2$ are exactly those of the form $A_r(y)^2$, a Pell family that includes $36$ and $1225$. Whether every square $N\ge4$ is representable for some $k$ remains open.

The formal-conjectures file also tags three other variants solved, and none needs its own page. The variant nine is a Lean witness ($k=3$, $n=11$, $m=25$) for $N=9$, added by pull request 4321 of 2026-06-25, whose description says it was drafted with Claude; a thread search of 2025-08-09 had already represented $N=9$. The variant four_two shows that $4$ is not representable with $k=2$, which settles no instance. The variant four_three is tagged solved but has no proof in the file, and it settles no instance.

Nat Sothanaphan linked three further notes in the thread, and none needs its own page. The note of 2026-03-04, generated by GPT-5.2 Thinking, is subsumed by the note of 2026-03-08: its representable squares, with roots congruent to $2$ mod $4$ or from the Pell families of its divisibility theorem, all have the form $A_r(y)^2$. The note of 2026-03-12, generated by GPT-5.4 Thinking, treats $k=3$; it represents $N=9$ and $N=16$, which a thread search had already represented on 2025-08-09, and its non-representation results for $k=3$ settle no instance. The note of 2026-03-14, generated by GPT-5.4 Thinking, excludes only $k$ in a finite set, so it settles no instance.