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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 distinctsettings: firstly, where $a$ and $b$ are both positive integers with $c$ and$d$ both Gaussian integers; secondly, where all of $a$, $b$, $c$, and $d$ areGaussian integers.

Gaussian integer solutions for the fifth power taxicab number problem