Primality proof for n = 2237:
Take b = 2.
b^(n-1) mod n = 1.
43 is prime.
b^((n-1)/43)-1 mod n = 1983, which is a unit, inverse 502.
13 is prime.
b^((n-1)/13)-1 mod n = 213, which is a unit, inverse 2216.
(13 * 43) divides n-1.
(13 * 43)^2 > n.
n is prime by Pocklington's theorem.