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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 expressedas the sum of two integer squares. We prove that $a^n + 1$ is the sum of twosquares for all $n \in \mathbb{N}$ if and only if $a$ is a perfect square. Wealso prove that for $a\equiv 0,1,2\pmod{4},$ if $a^{n} + 1$ is the sum of twosquares, then $a^{\delta} + 1$ is the sum of two squares for all $\delta | n, \\delta>1$. Using Aurifeuillian factorization, we show that if $a$ is a primeand $a\equiv 1 \pmod{4}$, then there are either zero or infinitely many odd $n$such that $a^n+1$ is the sum of two squares. When $a\equiv 3\pmod{4},$ wedefine $m$ to be the least positive integer such that $\frac{a+1}{m}$ is thesum of two squares, and prove that if $a^n+1$ is the sum of two squares for anyodd integer $n,$ then $m | n$, and both $a^m+1$ and $\frac{n}{m}$ are sums oftwo squares.

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