Primality proof for n = 172054593956031949258510691:

Take b = 2.

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

4434155615661930479 is prime.
b^((n-1)/4434155615661930479)-1 mod n = 81982237619807054769897706, which is a unit, inverse 154275350521713489305385904.

(4434155615661930479) divides n-1.

(4434155615661930479)^2 > n.

n is prime by Pocklington's theorem.