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

Leyda Almodovar; Alyssa N. Byrnes; Julie Fink; Xiao Guan; Aashita Kesarwani; Gary Lavigne; Victor H. Moll; Luis A. Medina; Isabelle Nogues; Eric Rowland; Amber Yuan

Vol. 29 (2019)

Articles

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

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Leyda Almodovar,
Alyssa N. Byrnes,
Julie Fink,
Xiao Guan,
Aashita Kesarwani,
Gary Lavigne,
Victor H. Moll,
Luis A. Medina,
Isabelle Nogues,
Eric Rowland,
Amber Yuan

more info

Leyda Almodovar
Department of Mathematics, University of Iowa , Iowa City, IA 52242, U.S.A.

Alyssa N. Byrnes
Department of Mathematics, Tulane University, New Orleans, U.S.A.

Julie Fink
330 Hudson St, 7th Floor, New York, NY 10013, U.S.A.

Xiao Guan
Department of Mathematics, Tulane University, New Orleans, U.S.A

Aashita Kesarwani
Department of Mathematics, Tulane University, New Orleans, U.S.A.

Gary Lavigne
Department of Mathematics, Tulane University, New Orleans, U.S.A

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

Luis A. Medina
Departamento de Matematicas, Universidad de Puerto Rico, Ri0 Piedras, San Juan, PR 00936-8377

Isabelle Nogues
e Department of Mathematics, Princeton University, Princeton, NJ 08544, U.S.A

Eric Rowland
Laboratoire de Combinatoire et d’informatique Math´ematique (LaCIM), Universite du Quebec a Montreal, CP 8888, Succ. Centre-ville, Montr ´ ´eal (Quebec) H3C 3P8, CANADA

Amber Yuan
g Department of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A.

Published 2024-09-04

Keywords

valuations,
quadratic polynomials,
congruences modulo 8

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.

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