Abstract
Using Fermat’s two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form can be expressed as the sum of two integer squares. We prove that is the sum of two squares for all if and only if is a square. We also prove that if , is odd, and is the sum of two squares, then is the sum of two squares for all , . Using Aurifeuillian factorization, we show that if is a prime and , then there are either zero or infinitely many odd such that is the sum of two squares. When , we define to be the least positive integer such that is the sum of two squares, and prove that if is the sum of two squares for odd, then , and both and are sums of two squares.
Citation
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. https://doi.org/10.2140/involve.2019.12.585
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