Primality proof for n = 90050068090664042551646936303486902724615855323314057604193773:

Take b = 2.

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

669909697361913545660898382546007419518544524592370543 is prime.
b^((n-1)/669909697361913545660898382546007419518544524592370543)-1 mod n = 69956533408516809605491217315586076635329733994132581285800710, which is a unit, inverse 61769177632366297338544823469855568429487710022487625510590641.

(669909697361913545660898382546007419518544524592370543) divides n-1.

(669909697361913545660898382546007419518544524592370543)^2 > n.

n is prime by Pocklington's theorem.