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Claim. Every non-square integer $N\ge2$ equals $\frac{(m+1)(m+2)}{(n+1)(n+2)}$ for some $n\ge0$ and $m\ge n+2$. So the representation that Problem 686 asks for exists with $k=2$ for every non-square $N$. The proof is the theorem erdos_686.variants.non_square of the formal-conjectures statement file for the problem. It was merged on 2 March 2026 from the pull request linked above, which was opened on 25 February 2026. The pull request's description says that it includes an AlphaProof formal proof, cleaned up for maintainability, with the raw output kept in the pull request's history. The proof settles $N=2$ and $N=3$ by a finite search checked with native_decide. It obtains every other case from a nontrivial solution of a Pell equation, which exists because $N$ is not a square. A thread post of 3 March 2026 links the pull request and names AlphaProof. A thread post of 9 August 2025 had already observed, without a general proof, that generalized Pell equations give such representations.

Covers. The question for every non-square $N\ge2$, answered yes with $k=2$. Not covered: square $N$, where the problem stays open.

Depends on. No page of this wiki.

Acceptance. None recorded. The site labels the problem OPEN, and its commentary does not mention the result. This corpus has not built the file at the linked commit or audited the theorem's statement, so the link is not formalized evidence. The claimant is the organization, and the system is named as the pull request names it.