Primality proof for n = 37447:

Take b = 2.

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

79 is prime.
b^((n-1)/79)-1 mod n = 17193, which is a unit, inverse 26082.

(79^2) divides n-1.

(79^2)^2 > n.

n is prime by Pocklington's theorem.