Primality proof for n = 37663:

Take b = 2.

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

6277 is prime.
b^((n-1)/6277)-1 mod n = 63, which is a unit, inverse 13750.

(6277) divides n-1.

(6277)^2 > n.

n is prime by Pocklington's theorem.