Primality proof for n = 45677:

Take b = 2.

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

601 is prime.
b^((n-1)/601)-1 mod n = 4545, which is a unit, inverse 18311.

(601) divides n-1.

(601)^2 > n.

n is prime by Pocklington's theorem.