Primality proof for n = 1779835204363103:

Take b = 2.

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

889917602181551 is prime.
b^((n-1)/889917602181551)-1 mod n = 3, which is a unit, inverse 593278401454368.

(889917602181551) divides n-1.

(889917602181551)^2 > n.

n is prime by Pocklington's theorem.