Primality proof for n = 123127:

Take b = 2.

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

20521 is prime.
b^((n-1)/20521)-1 mod n = 63, which is a unit, inverse 9772.

(20521) divides n-1.

(20521)^2 > n.

n is prime by Pocklington's theorem.