Primality proof for n = 38116200399806648171588966205760165042098583043261:

Take b = 2.

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

209868713262055181656334059536197675522401 is prime.
b^((n-1)/209868713262055181656334059536197675522401)-1 mod n = 13008256837837504966483997409530291437166825213693, which is a unit, inverse 11603348861056018800355564817618637950191603546405.

(209868713262055181656334059536197675522401) divides n-1.

(209868713262055181656334059536197675522401)^2 > n.

n is prime by Pocklington's theorem.