Primality proof for n = 1937551337942089:

Take b = 2.

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

213095137 is prime.
b^((n-1)/213095137)-1 mod n = 1240887883484574, which is a unit, inverse 1427436816863515.

(213095137) divides n-1.

(213095137)^2 > n.

n is prime by Pocklington's theorem.