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.