## Involve: A Journal of Mathematics

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

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

#### 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 $n∈ℕ$ 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 $n∕m$ are sums of two squares.

#### Article information

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

Dates
Revised: 20 June 2018
Accepted: 24 June 2018
First available in Project Euclid: 30 May 2019

https://projecteuclid.org/euclid.involve/1559181654

Digital Object Identifier
doi:10.2140/involve.2019.12.585

Mathematical Reviews number (MathSciNet)
MR3941600

Zentralblatt MATH identifier
07072541

#### Citation

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. https://projecteuclid.org/euclid.involve/1559181654