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

Olena Kozhushkina; Maila Brucal Hallare; Jane Long; Victor H. Moll; Jean-Claude Pedjeu; Bianca  Thompson; Justin Trulen

Vol. 30 (2020)

Articles

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

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Olena Kozhushkina,
Maila Brucal Hallare,
Jane Long,
Victor H. Moll,
Jean-Claude Pedjeu,
Bianca Thompson,
Justin Trulen

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Olena Kozhushkina
Department of Mathematics and Computer Science, Ursinus College, Collegeville, PA 19426

Maila Brucal Hallare
Department of Mathematics, Norfolk State University, Norfolk, VA 23504

Jane Long
Department of Mathematics and Statistics, Stephen F. Austin State University, P.O. Box 13040, SFA Station, Nacogdoches, TX 75962-3040

Victor H. Moll
Department of Mathematics, Tulane University, New Orleans, LA 70118, U.S.A.

Jean-Claude Pedjeu
Department of Mathematics, Tennessee State University, Nashville, TN 37209

Bianca Thompson
Department of Mathematics, Westmister College, Salt Lake City, UT 84105.

Justin Trulen
Department of Mathematics, Division of Natural Sciences and Mathematics, Kentucky Wesleyan College, Owensboro, KY 42301

Published 11-07-2024

Keywords

quadratic polynomials

Abstract

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.

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