Primality proof for n = 3378892281805798826440512057402586184339:

Take b = 2.

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

3752823319668658282198008768437 is prime.
b^((n-1)/3752823319668658282198008768437)-1 mod n = 184946080192435110328219018408549828350, which is a unit, inverse 659624996574668187614555563240690487199.

(3752823319668658282198008768437) divides n-1.

(3752823319668658282198008768437)^2 > n.

n is prime by Pocklington's theorem.