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

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?

• It's rare for a random sum to be prime. If you found that $a^n+b^n$ is often prime for some other $a$ and $b$, that would be a more interesting thing to look at. Can you say more about "a lot more than not that for a^4+b^4, their sums made a prime number"? Dec 25, 2019 at 7:16
• For odd $n$ it cannot be a prime because $5^n+12^n \equiv 0 \pmod{17}$. Dec 25, 2019 at 7:16
• Also, if $n$ is $2$ times an odd integer, then $5^n + 12^n \equiv 0 \pmod{13}$. Dec 25, 2019 at 7:21
• Apparently $5^{16}+12^{16}$ is a prime number. Dec 25, 2019 at 7:23

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.