Does $5^n + 12^n$ = a prime number for any $n>2$? Why or why not?

Question

I was randomly checking on what numbers come out of taking the A and B terms of pythagorean triples and adding them to powers higher than two on a ti84+ to sleep, and I noticed a lot more than not that for $a^4+b^4$, their sums made a prime number. The one odd exception is $5^n+12^n$, but not only that, when I checked up to $n=14$ on https://www.numberempire.com/numberfactorizer.php, $5^n+12^n$ still wasn't prime...Not even random higher values of $n=43$ and $n=50$ were prime...Is this already a known "thing" in math, that certain A and B terms in pythagorean triples can sum to prime numbers at powers greater than 2, while for pairs like 5 and 12, no primes can be made? If so, is there at least an publicly accessible explanation, if not a simple and/or intuitive one?

Answer 1

This is generally not a type of question worth pursuing. That being said, $5^{16}+12^{16}=184884411482927041$ is a prime.

Of course this could only happen when $n$ is a power of 2. Otherwise $n=m\cdot k$ where $k$ is odd, and then $a^n+b^n=(a^m)^k+(b^m)^k$, which is divisible by $a^m+b^m$. There was no point in checking 43 and 50.

With double exponential function growing very fast, you can only check a handful of numbers, so no wonder if for some $a$ and $b$ none of these is prime. This is not a fact of any consequence. We don't know if there are infinitely many primes of this form for any specific $a$ and $b$, nor do we know whether there is at least one such prime. We got lucky with $(a,b)=(5,12)$, and we got lucky a few times with $(a,b)=(1,2)$ (see Fermat primes), but that's about it.