Primality proof for n = 124822877:

Take b = 2.

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

4273 is prime.
b^((n-1)/4273)-1 mod n = 122492350, which is a unit, inverse 35056210.

109 is prime.
b^((n-1)/109)-1 mod n = 88257422, which is a unit, inverse 7603136.

(109 * 4273) divides n-1.

(109 * 4273)^2 > n.

n is prime by Pocklington's theorem.