When is an + 1 the sum of two squares ?

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

doi:10.2140/involve.2019.12.585

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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 $a^{n} + 1$ can be expressed as the sum of two integer squares. We prove that $a^{n} + 1$ is the sum of two squares for all $n \in ℕ$ if and only if $a$ is a square. We also prove that if $a \equiv 0, 1, 2 (mod 4)$ , $n$ is odd, and $a^{n} + 1$ is the sum of two squares, then $a^{δ} + 1$ is the sum of two squares for all $δ | n$ , $δ > 1$ . Using Aurifeuillian factorization, we show that if $a$ is a prime and $a \equiv 1 (mod 4)$ , then there are either zero or infinitely many odd $n$ such that $a^{n} + 1$ is the sum of two squares. When $a \equiv 3 (mod 4)$ , we define $m$ to be the least positive integer such that $(a + 1) ∕ m$ is the sum of two squares, and prove that if $a^{n} + 1$ is the sum of two squares for $n$ odd, then $m | n$ , and both $a^{m} + 1$ and $n ∕ m$ are sums of two squares.

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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

Vol. 12, No. 4, 2019