The valuation tree for \( n² + 7\)

2025/09/09 by Kozhushkina, Olena, Hallare, Maila Brucal, Long, Jane +3
#FOS: Mathematics #quadratic polynomials

paper · doi:10.71712/adsc-sw47

Abstract

The 2-adic valuation of an integer \( x\) is the highest power of 2 which divides \( x\) It is denoted by \( v₂(x).\) The goal of the present work is to describe the sequence \(\v₂(n² + a)\\) for \( 1 ≤ a ≤ 7\)\)  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.

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