Primality proof for n = 126241:

Take b = 2.

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

263 is prime.
b^((n-1)/263)-1 mod n = 40022, which is a unit, inverse 81428.

5 is prime.
b^((n-1)/5)-1 mod n = 118277, which is a unit, inverse 71284.

(5 * 263) divides n-1.

(5 * 263)^2 > n.

n is prime by Pocklington's theorem.