Primality proof for n = 142211:

Take b = 2.

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

14221 is prime.
b^((n-1)/14221)-1 mod n = 1023, which is a unit, inverse 10148.

(14221) divides n-1.

(14221)^2 > n.

n is prime by Pocklington's theorem.