Primality proof for n = 2585077427327:

Take b = 2.

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

76031689039 is prime.
b^((n-1)/76031689039)-1 mod n = 17179869183, which is a unit, inverse 2050746774463.

(76031689039) divides n-1.

(76031689039)^2 > n.

n is prime by Pocklington's theorem.