Primality proof for n = 10141:

Take b = 2.

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

13 is prime.
b^((n-1)/13)-1 mod n = 598, which is a unit, inverse 5274.

(13^2) divides n-1.

(13^2)^2 > n.

n is prime by Pocklington's theorem.