VibeMathedMath problems solved by AI

Amdeberhan-Medina-Moll Arctangent Sum Conjecture

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.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Number theory
Posed by
Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll
Year posed
2008
Years open
18y
Solved
2026-07-07
Model
AxiomProver
Vendor
Axiom Math
Collaborators
Ken Ono
Verification
Unreviewed
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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

What the AI did

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.

Verification

A public Lean 4.28.0 development accompanies the paper, containing a formalization of the problem and a proof the author states is sorry-free and adds no axioms. We attempted to compile it and did not complete the build, so the axiom claim here rests on the author's statement rather than on our own check.

Sources

arXiv:2607.05739 - Integer values of tan(arctan 1 + arctan 2 + ... + arctan n) are rare

Discussion