Is $88888888^8 + 1$ prime?

Question

Background.

I was looking at Pythagorean primes $p= x^2 + y^2$, then at a subset, Quartan primes, $p= x^4 + y^4$ http://oeis.org/A002645

I wondered if there were primes of the form $p= x^8 + y^8$, but my searches only produced one result $88888888^8 + 1$ https://primes.utm.edu/curios/page.php/8.html

To cut a long story short, I put together some code to look for $p= x^8 + y^8$, and produced $(x,y,p)=$

$$(1,1,2)$$ $$(2,1,257)$$ $$(4,1,65537)$$ $$(6,5,2070241)$$ $$(10,3,100006561)$$ $$(12,7,435746497)$$ $$(13,2,815730977)$$ $$(13,8,832507937)$$

With these values, I found these were Octavan primes https://oeis.org/A006686

By my calculation, $$88888888^8 + 1^8=$$ $$3897443119493995135240117470484161627805761928038551890318852097$$

My question: Is this number prime?

Also, nice to know: Is there an online method to test numbers, of the order $4E+63$, for primality? Is there a compressive list of prime varieties?

Answer 1

Yes, your number is prime. I verified this first using an implementation of the Miller-Rabin probabilistic prime test, followed by the AKS primality test.

Interestingly, it turns out that WolframAlpha is capable of verifying that this is prime as well.