A closed-form solution might be given by a tree. Valuations of quadratic polynomials

Abstract

The \(p-\)adic valuation of an integer \(x\) is the largest power of the prime \(p\) that divides \(x\) . It is denoted by \(\nu_p(x)\). This work describes properties of the valuation \(\nu_2(n^2 + a)\), with \(a \in \mathbb{N}\) . A distinction of the behavior of these valuations for a \(\equiv 7 \pmod{8}\) or not is presented.