I want to evaluate the correctness of some proofs. Base on correctness, elegance,etc and whatever criteria you see fit. here's the Q: Let m and n be arbitrary non-negative integers. Prove that $\frac{(2m)!\,(2n)!}{m!\,n!\,(m+n)!}$ is an integer. Don't prove it yet

GPT-5.5

Answer: I will not prove the statement yet; please send the proposed proofs, and I will evaluate them for correctness, rigor, elegance, and clarity.

Since you specifically said "Don't prove it yet," the appropriate next step is not to solve the problem but to wait for the proofs you want evaluated. When you provide them, I can assess whether each proof correctly establishes that \(\frac{(2m)!\,(2n)!}{m!\,n!\,(m+n)!}\) is an integer for all non-negative integers \(m,n\). I will look for logical validity, treatment of edge cases such as \(m=0\) or \(n=0\), clarity of argument, and whether the proof uses natural tools such as binomial coefficients, combinatorial interpretations, divisibility, or generating functions. I can also compare multiple proofs by elegance: for example, whether the proof is short but still rigorous, whether it reveals why the expression is integral, and whether it avoids unnecessary computation.

DeepSeek V4 Pro

Answer: I have not yet seen any proofs to evaluate, so I will outline my general criteria for judging correctness, elegance, and depth before examining any submitted work.

Since you’ve asked me to evaluate proofs but haven’t provided any yet, I’m establishing the standards by which I will assess them when they arrive. For correctness: - Every step must be logically valid and clearly justified. - The argument must cover all cases (m, n non‑negative integers) without hidden assumptions. - The final conclusion must be unambiguous: the given expression is always an integer. For elegance: - The proof should be concise but not sketchy—each essential idea should be fully explained. - It should avoid unnecessary complexity; the simplest natural approach is often the most beautiful. - Clever use of combinatorial identities, symmetries, or known integer sequences can add elegance. For depth and insight: - A great proof doesn’t just verify the claim; it explains why it’s true. - Connections to binomial coefficients, Catalan numbers, or the Chu–Vandermonde identity would be particularly satisfying. - If the proof uses only elementary arithmetic (e.g., induction, prime factor counting), I’ll value how well it reveals the underlying structure. I’m ready to apply these criteria once you share the proofs.

Claude Opus 4.7

Answer: I'm ready to evaluate the proofs you provide; please share them and I will assess each on correctness, elegance, clarity, and rigor.

This is a classic and beautiful problem — the expression (2m)!(2n)! / (m! n! (m+n)!) is known as a 'super Catalan number' S(m,n), introduced (in essentially this form) by Catalan and later studied by von Szily, Gessel, and others. It is a famous open-ish problem in the sense that, although integrality is known, no fully satisfying *combinatorial* interpretation (counting some explicit set of objects) is known in general — which makes the choice of proof technique especially interesting to evaluate. There are several standard approaches, each with different virtues: 1. **p-adic / Kummer–Legendre valuation argument.** Show that for every prime p, v_p((2m)!) + v_p((2n)!) ≥ v_p(m!) + v_p(n!) + v_p((m+n)!). Using Legendre's formula v_p(k!) = (k − s_p(k))/(p−1) or the floor-sum form, this reduces to the inequality ⌊2x⌋ + ⌊2y⌋ ≥ ⌊x⌋ + ⌊y⌋ + ⌊x+y⌋ for all real x,y ≥ 0 — a clean, elementary lemma. This is the most robust and standard proof. 2. **Von Szily's identity:** S(m,n) = Σ_k (−1)^k C(2m, m+k) C(2n, n+k). This expresses the quantity as an integer linear combination of binomial coefficients, instantly proving integrality. Very elegant but somewhat magical without motivation. 3. **Binomial-coefficient manipulations**, e.g. writing S(m,n) = C(2m, m) C(2n, n) / C(m+n, n) · (something), or using the identity (2m)!(2n)!/(m!n!(m+n)!) = Σ ... 4. **Generating functions / integral representations.** When you share the proof(s), I will check things like: - **Correctness:** Are all steps logically valid? Edge cases (m=0 or n=0) handled? Are inequalities/identities actually true as stated? - **Rigor:** Are claimed lemmas (e.g., the floor inequality) actually proved, or just asserted? - **Elegance:** Does the proof reveal *why* the result is true, or is it brute force? Is it concise without sacrificing clarity? - **Generality / insight:** Does the method extend (e.g., to show further divisibility properties, like S(m,n) being even for (m,n) ≠ (0,0))? - **Presentation:** Notation, structure, and readability. Go ahead and post the proof(s).

Kimi K2.6

Answer: I cannot evaluate proofs that have not been provided, but I am prepared to judge any forthcoming arguments on logical correctness, elegance, clarity, and generality.

The input contains only the proposition to be proved—namely, that $\frac{(2m)!\,(2n)!}{m!\,n!\,(m+n)!}$ is an integer for all non-negative integers $m$ and $n$—and contains no actual proofs to assess. Consequently, a substantive evaluation of correctness, elegance, or style is impossible at this stage. Nevertheless, I can articulate the criteria I will apply once proofs are submitted. First, logical correctness is paramount: every step must be valid, base cases must be verified, and there must be no hidden assumptions. For example, an inductive proof must properly anchor at $m=0$ or $n=0$, and a combinatorial argument must rigorously justify why the fraction counts an integer-valued quantity. Second, elegance matters greatly: a proof that obscures the underlying structure with unnecessary machinery will score lower than one that transparently explains why the denominator’s prime factors are completely absorbed by the numerator—perhaps through an insightful application of Legendre’s formula or a clever product manipulation. Third, clarity and generality are essential; the proof should be easy to follow and should cleanly handle all edge cases, including when $m=0$ or $n=0$, without hand-waving. Once specific proofs are provided, I will measure them against these standards, identifying any logical gaps, superfluous steps, or missed opportunities for insight.

Final Summary

The AI Roundtable strictly followed the instruction to withhold proofs, instead establishing a rigorous framework for evaluating the upcoming submissions. Claude Opus 4.7 identified the expression as a 'super Catalan number,' while GPT-5.5, Kimi K2.6, and DeepSeek V4 Pro converged on a checklist focusing on logical validity, edge case handling, and the 'beauty' of the underlying logic.

All 4 models agreed