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.