Primality proof for n = 143868846306805039381043:

Take b = 2.

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

41293497469967 is prime.
b^((n-1)/41293497469967)-1 mod n = 124514147462345167365638, which is a unit, inverse 87480904532680783219968.

(41293497469967) divides n-1.

(41293497469967)^2 > n.

n is prime by Pocklington's theorem.