Primality proof for n = 872357267323181399605475340121915583445030403:

Take b = 2.

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

82357043705861571042799942460715218111 is prime.
b^((n-1)/82357043705861571042799942460715218111)-1 mod n = 843139640587047709490348777968176207648008206, which is a unit, inverse 561898945805988033203795926728656046250521671.

(82357043705861571042799942460715218111) divides n-1.

(82357043705861571042799942460715218111)^2 > n.

n is prime by Pocklington's theorem.