The valuation tree for \(\ n^2 + 7\)

The 2-adic valuation of an integer \( x\) is the highest power of 2 which divides \( x\) It is denoted by \(\ v_2(x).\) The goal of the present work is to describe the sequence \(\{v_2(n^2 + a)\}\) for \(\ 1 \leq a \leq 7\)\) The first six cases are elementary. The last case considered here, namely \(\ a = 7\), presents distinct challenges. It is shown here how to represent this family of valuations in the form of an infinite binary tree, with two symmetric infinite branches.