Primality proof for n = 79879924901292653287162734017:

Take b = 2.

b^(n-1) mod n = 1.

28591973599411 is prime.
b^((n-1)/28591973599411)-1 mod n = 25873039023136504529196578628, which is a unit, inverse 73292942410362244624544064009.

3086977169 is prime.
b^((n-1)/3086977169)-1 mod n = 9655117697654085619936833249, which is a unit, inverse 15463752343466024101614486223.

(3086977169 * 28591973599411) divides n-1.

(3086977169 * 28591973599411)^2 > n.

n is prime by Pocklington's theorem.