Primality proof for n = 25339:

Take b = 2.

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

103 is prime.
b^((n-1)/103)-1 mod n = 21399, which is a unit, inverse 13885.

41 is prime.
b^((n-1)/41)-1 mod n = 1860, which is a unit, inverse 22628.

(41 * 103) divides n-1.

(41 * 103)^2 > n.

n is prime by Pocklington's theorem.