Primality proof for n = 120699720968197491947347:

Take b = 2.

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

290064143 is prime.
b^((n-1)/290064143)-1 mod n = 22990475459624712861577, which is a unit, inverse 26729462098473212341540.

154950581 is prime.
b^((n-1)/154950581)-1 mod n = 73072935989190308070127, which is a unit, inverse 11720905197638924089857.

(154950581 * 290064143) divides n-1.

(154950581 * 290064143)^2 > n.

n is prime by Pocklington's theorem.