TITLE: Is $88888888^8 + 1$ prime?
QUESTION [8 upvotes]: 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?

REPLY [4 votes]: 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.