Primality proof for n = 7523:

Take b = 2.

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

3761 is prime.
b^((n-1)/3761)-1 mod n = 3, which is a unit, inverse 2508.

(3761) divides n-1.

(3761)^2 > n.

n is prime by Pocklington's theorem.