Integer values of $\tan(\arctan 1+\arctan 2+\cdots+\arctan n)$ are rare

Abstract

For $n\ge1$, we let $$x_n:=\tan\bigl(\sum_{k=1}^{n}\arctan k\bigr).$$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_n\not \in \mathbb{Z}$ for every $n\ge5$. This was known for a set of positive integers of density $\tfrac{120}{817}\approx0.1469$. We prove that an integer value $x_n=m$ satisfies $|m|\ge e^{(1/2+o(1))\,n\log n}$, which we use to deduce that $$\#\{\,1\leq n\le N:x_n\in\mathbb{Z}\,\}=O(\log N). $$ In particular, the conjecture holds for a density-one set of $n$. The results in this note were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.