VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Let $x_n = \tan\left(\sum_{k=1}^{n} \arctan k\right)$. Amdeberhan, Medina and Moll conjectured that $x_n \notin \mathbb{Z}$ for every $n \ge 5$. Any integer value $x_n = m$ must satisfy $|m| \ge e^{(1/2+o(1)) n \log n}$, which forces $\#\{1 \le n \le N : x_n \in \mathbb{Z}\} = O(\log N)$. The conjecture therefore holds for a density-one set of $n$, improving on the previously known density of $120/817 \approx 0.147$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
AxiomProver
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.
Provider: Axiom Math · Prompt public: unknown
· Independence: unknown
Verification boundary
unreviewed
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.