The valuation tree for \(\ n^2 + 7\) | Scientia Series A: Mathematical Sciences

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Published: 2020-01-08

DOI: https://doi.org/10.71712/adsc-sw47

Keywords:

quadratic polynomials

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, Westminster College, Salt Lake City, UT 84105.

https://orcid.org/0000-0002-0324-2190 (unauthenticated)

Justin Trulen

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

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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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

How to Cite

Kozhushkina, O., Hallare, M. B., Jane Long, Moll, V. H., Pedjeu, J.-C., Thompson, B., & Trulen, J. (2020). The valuation tree for \(\ n^2 + 7\). Scientia Series A: Mathematical Sciences, 30, 91-102. https://doi.org/10.71712/adsc-sw47

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