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2019 When is $a^{n} + 1$ the sum of two squares?
Greg Dresden, Kylie Hess, Saimon Islam, Jeremy Rouse, Aaron Schmitt, Emily Stamm, Terrin Warren, Pan Yue
Involve 12(4): 585-605 (2019). DOI: 10.2140/involve.2019.12.585


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 a0,1,2(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 a1(mod4), then there are either zero or infinitely many odd n such that an+1 is the sum of two squares. When a3(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.


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Greg Dresden. Kylie Hess. Saimon Islam. Jeremy Rouse. Aaron Schmitt. Emily Stamm. Terrin Warren. Pan Yue. "When is $a^{n} + 1$ the sum of two squares?." Involve 12 (4) 585 - 605, 2019.


Received: 11 October 2017; Revised: 20 June 2018; Accepted: 24 June 2018; Published: 2019
First available in Project Euclid: 30 May 2019

zbMATH: 07072541
MathSciNet: MR3941600
Digital Object Identifier: 10.2140/involve.2019.12.585

Primary: 11E25
Secondary: 11C08 , 11R18

Keywords: cyclotomic polynomials , Fermat's two squares theorem

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.12 • No. 4 • 2019
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