I saw a question a while back which asked (paraphrased): how many examples exist of the form $a^n + b^n = c^2$ where $b$ is $a$'s digits in reverse, $n \geq 2$, and $a,b,c$ are coprime.
For $n = 2$, the proof is not too hard that there are zero solutions (via mod 3).
For $n \geq 4$, the reverse part isn't even needed, there are no solutions (http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8631.pdf contains a proof).
For $n = 3$, $56^3 + 65^3$ is a square. My question is whether there exists more cases. Trying mod 9 gives some restrictions, but I couldn't narrow it that significantly. Coding a quick test shows there are no other "small solutions" (Up to a = 10 million). Code viewable here: https://codeinterview.io/YRGDWJVCAH