Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 4 (2019), 585-605.

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

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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.

Article information

Involve, Volume 12, Number 4 (2019), 585-605.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 11C08: Polynomials [See also 13F20] 11R18: Cyclotomic extensions

cyclotomic polynomials Fermat's two squares theorem


Dresden, Greg; Hess, Kylie; Islam, Saimon; Rouse, Jeremy; Schmitt, Aaron; Stamm, Emily; Warren, Terrin; Yue, Pan. When is $a^{n} + 1$ the sum of two squares?. Involve 12 (2019), no. 4, 585--605. doi:10.2140/involve.2019.12.585.

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