  The famous open problem of finding positive integer solutions to $a^5 + b^5 =
c^5 + d^5$ is considered, and related solutions are found in two distinct
settings: firstly, where $a$ and $b$ are both positive integers with $c$ and
$d$ both Gaussian integers; secondly, where all of $a$, $b$, $c$, and $d$ are
Gaussian integers.
