Primality proof for n = 168650669431260323389:

Take b = 2.

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

12695774573265607 is prime.
b^((n-1)/12695774573265607)-1 mod n = 498220274880388418, which is a unit, inverse 132237738639215230653.

(12695774573265607) divides n-1.

(12695774573265607)^2 > n.

n is prime by Pocklington's theorem.