Primality proof for n = 19197701325821:

Take b = 2.

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

41734133317 is prime.
b^((n-1)/41734133317)-1 mod n = 17447704774951, which is a unit, inverse 17507678135042.

(41734133317) divides n-1.

(41734133317)^2 > n.

n is prime by Pocklington's theorem.