Primality proof for n = 34646440928557194402992574983797:

Take b = 2.

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

136401162692544977256234449 is prime.
b^((n-1)/136401162692544977256234449)-1 mod n = 17964283499622987307244987545888, which is a unit, inverse 5037174433048681144301386838262.

(136401162692544977256234449) divides n-1.

(136401162692544977256234449)^2 > n.

n is prime by Pocklington's theorem.