Primality proof for n = 9361973132609:

Take b = 3.

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

105465631 is prime.
b^((n-1)/105465631)-1 mod n = 8513967533808, which is a unit, inverse 7570366960336.

(105465631) divides n-1.

(105465631)^2 > n.

n is prime by Pocklington's theorem.