Primality proof for n = 7753:

Take b = 2.

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

19 is prime.
b^((n-1)/19)-1 mod n = 2575, which is a unit, inverse 7476.

17 is prime.
b^((n-1)/17)-1 mod n = 945, which is a unit, inverse 6588.

(17 * 19) divides n-1.

(17 * 19)^2 > n.

n is prime by Pocklington's theorem.