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Amdeberhan-Medina-Moll Arctangent Sum Conjecture

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amdeberhan-medina-moll-arctan-conjectureNumber theoryposed by Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll, 2008recorded: partial

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Statement

Let xn=tan⁡(∑k=1narctan⁡k)x_n = \tan\left(\sum_{k=1}^{n} \arctan k\right). Amdeberhan, Medina and Moll conjectured that xn∉Zx_n \notin \mathbb{Z} for every n≥5n \ge 5. Any integer value xn=mx_n = m must satisfy ∣m∣≥e(1/2+o(1))nlog⁡n|m| \ge e^{(1/2+o(1)) n \log n}, which forces #{1≤n≤N:xn∈Z}=O(log⁡N)\#\{1 \le n \le N : x_n \in \mathbb{Z}\} = O(\log N). The conjecture therefore holds for a density-one set of nn, improving on the previously known density of 120/817≈0.147120/817 \approx 0.147.

Context

density-one set of n; the conjecture itself remains open

A named conjecture from a 2008 Journal of Number Theory paper with a documented line of partial results, familiar within the arctangent-sums and Gaussian-integer literature but not beyond it.

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  • #1

    Attempt 1

    proof attemptAxiomProver with Ken Ono ·
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    — the result was found by a model.
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    The system was given only a natural-language statement of the definitions and the three target results, plus an instruction to formalize and prove them with no sorry. From that input AxiomProver autonomously produced both the Lean formalization of the problem and a complete Lean proof. The human author then wrote the paper's exposition using the formal development as his reference, which reverses the usual order: the Lean came first and the prose was derived from it.

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