An arithmetic conjecture on an arctangent sum - Victor H. Moll (Scientia, 2025)
A sequence \(x_n\), defined in terms of a sum of arctangent values, satisfies the nonlinear recurrence \(x_n = (n + x_{n-1})/(1 - nx_{n-1})\), with \(x_1 = 1\), which has been conjectured not to be an integer for \(n ≥ 5\). This problem is analyzed here in terms of divisibility questions of an associated sequence. Properties of this new sequence are employed to prove that the subsequences \(\{x_{19n+5} : n \in \mathbb{N}\}\) and \(\{x_{19n+13} : n \in \mathbb{N}\}\) contain no integer values.