Primality proof for n = 44959:

Take b = 2.

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

127 is prime.
b^((n-1)/127)-1 mod n = 27225, which is a unit, inverse 29000.

59 is prime.
b^((n-1)/59)-1 mod n = 30728, which is a unit, inverse 10337.

(59 * 127) divides n-1.

(59 * 127)^2 > n.

n is prime by Pocklington's theorem.