Primality proof for n = 174723607534414371449:

Take b = 2.

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

44706919 is prime.
b^((n-1)/44706919)-1 mod n = 69188669156729060296, which is a unit, inverse 97513711501333925610.

120233 is prime.
b^((n-1)/120233)-1 mod n = 97784642313376440035, which is a unit, inverse 14283963398578376782.

(120233 * 44706919) divides n-1.

(120233 * 44706919)^2 > n.

n is prime by Pocklington's theorem.