Primality proof for n = 132401351950596217829363773663:

Take b = 2.

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

25910585072550149 is prime.
b^((n-1)/25910585072550149)-1 mod n = 49074710227844446254173682114, which is a unit, inverse 8308257894070757136273938585.

(25910585072550149) divides n-1.

(25910585072550149)^2 > n.

n is prime by Pocklington's theorem.