Vol. 12, No. 4, 2019

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When is $a^{n} + 1$ the sum of two squares?

Greg Dresden, Kylie Hess, Saimon Islam, Jeremy Rouse, Aaron Schmitt, Emily Stamm, Terrin Warren and Pan Yue

Vol. 12 (2019), No. 4, 585–605
DOI: 10.2140/involve.2019.12.585
Abstract

Using Fermat’s two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form an + 1 can be expressed as the sum of two integer squares. We prove that an + 1 is the sum of two squares for all n if and only if a is a square. We also prove that if a 0,1,2(mod4), n is odd, and an + 1 is the sum of two squares, then aδ + 1 is the sum of two squares for all δ|n, δ > 1. Using Aurifeuillian factorization, we show that if a is a prime and a 1(mod4), then there are either zero or infinitely many odd n such that an + 1 is the sum of two squares. When a 3(mod4), we define m to be the least positive integer such that (a + 1)m is the sum of two squares, and prove that if an + 1 is the sum of two squares for n odd, then m|n, and both am + 1 and nm are sums of two squares.

Keywords
cyclotomic polynomials, Fermat's two squares theorem
Mathematical Subject Classification 2010
Primary: 11E25
Secondary: 11C08, 11R18
Milestones
Received: 11 October 2017
Revised: 20 June 2018
Accepted: 24 June 2018
Published: 16 April 2019

Communicated by Kenneth S. Berenhaut
Authors
Greg Dresden
Department of Mathematics
Washington and Lee University
Lexingston, VA
United States
Kylie Hess
Department of Mathematics and Computer Science
Emory University
Atlanta, GA
United States
Saimon Islam
Department of Mathematics
Washington and Lee University
Lexington, VA
United States
Jeremy Rouse
Department of Mathematics and Statistics
Wake Forest University
Winston-Salem, NC
United States
Aaron Schmitt
Department of Mathematics
Washington and Lee University
Lexington, VA
United States
Emily Stamm
Department of Mathematics and Statistics
Vassar College
Poughkeepsie, NY
United States
Terrin Warren
Department of Mathematics
University of Georgia
Athens, GA
United States
Pan Yue
Department of Mathematics
Washington and Lee University
Lexington, VA
United States