Primality proof for n = 104719073621178708975837602950775180438320278101:

Take b = 2.

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

203852586375664218368381551393371968928013 is prime.
b^((n-1)/203852586375664218368381551393371968928013)-1 mod n = 40201969824907835965098388450003519734389074470, which is a unit, inverse 90084448839335833881019754118450260906209787307.

(203852586375664218368381551393371968928013) divides n-1.

(203852586375664218368381551393371968928013)^2 > n.

n is prime by Pocklington's theorem.