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 ${a}^{n}+1$ 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 $a\equiv 0,1,2\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$, $n$ is odd, and ${a}^{n}+1$ is the sum of two squares, then ${a}^{\delta }+1$ is the sum of two squares for all $\delta \phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}n$, $\delta >1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\equiv 1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$, then there are either zero or infinitely many odd $n$ such that ${a}^{n}+1$ is the sum of two squares. When $a\equiv 3\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}4\right)$, we define $m$ to be the least positive integer such that $\left(a+1\right)∕m$ is the sum of two squares, and prove that if ${a}^{n}+1$ is the sum of two squares for $n$ odd, then $m\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}n$, and both ${a}^{m}+1$ and $n∕m$ 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