Primality proof for n = 7757:

Take b = 2.

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

277 is prime.
b^((n-1)/277)-1 mod n = 4470, which is a unit, inverse 6780.

(277) divides n-1.

(277)^2 > n.

n is prime by Pocklington's theorem.