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The Tu-Deng Conjecture

Combinatorics · posed by Ziran Tu, Yingpu Deng, 2011 · solved

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Statement

With N=2k−1N = 2^k - 1 and wt(n)\mathrm{wt}(n) the binary Hamming weight, Tu and Deng conjectured that for every 1≤t≤N−11 \leq t \leq N-1 at most 2k−12^{k-1} pairs (a,b)(a,b) satisfy a+b≡t(modN)a + b \equiv t \pmod N and wt(a)+wt(b)<k\mathrm{wt}(a) + \mathrm{wt}(b) < k. Proved in full.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Renzhang Liu, Hengyi Luo and Tianyuan Xie, using ChatGPT 5.6 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai assisted
    — a person led the work and used a model along the way.

    "The authors thank ChatGPT 5.6 Pro for assistance with some of the mathematical work presented in this paper. The authors subsequently verified the argument and take full responsibility for the final content." No individual step is attributed, so the lower tier applies.

  2. Machine-checked by Lean on #1 · not a person

    lean: partially checkedLean

    scope Lean formalization of the core argument; statement correspondence not independently audited

    The authors provide an accompanying Lean formalization described as an end-to-end machine-checked proof, including the intermediate results and the reduction to the original statement. Nobody independent has audited whether the formal statement faithfully expresses the conjecture, so this sits on the unaudited Lean rung.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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