Primality proof for n = 1191729595807457796514621337:

Take b = 2.

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

19197701325821 is prime.
b^((n-1)/19197701325821)-1 mod n = 50353361797821131950438234, which is a unit, inverse 998934475662477691726286740.

5080793 is prime.
b^((n-1)/5080793)-1 mod n = 233069766110672735520977925, which is a unit, inverse 767223022792207964435388427.

(5080793 * 19197701325821) divides n-1.

(5080793 * 19197701325821)^2 > n.

n is prime by Pocklington's theorem.