Primality proof for n = 669909697361913545660898382546007419518544524592370543:

Take b = 2.

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

2788900573532341797131599550190476443808598899 is prime.
b^((n-1)/2788900573532341797131599550190476443808598899)-1 mod n = 640093040736131505187168390086969230074145600489316288, which is a unit, inverse 519136795617045121268162467342718511295987927661735653.

(2788900573532341797131599550190476443808598899) divides n-1.

(2788900573532341797131599550190476443808598899)^2 > n.

n is prime by Pocklington's theorem.