Primality proof for n = 3137:
Take b = 3.
b^(n-1) mod n = 1.
7 is prime.
b^((n-1)/7)-1 mod n = 2662, which is a unit, inverse 317.
2 is prime.
b^((n-1)/2)-1 mod n = 3135, which is a unit, inverse 1568.
(2^6 * 7^2) divides n-1.
(2^6 * 7^2)^2 > n.
n is prime by Pocklington's theorem.