Primality proof for n = 85253989:

Take b = 2.

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

1879 is prime.
b^((n-1)/1879)-1 mod n = 21907381, which is a unit, inverse 55718753.

199 is prime.
b^((n-1)/199)-1 mod n = 34308376, which is a unit, inverse 41746173.

(199 * 1879) divides n-1.

(199 * 1879)^2 > n.

n is prime by Pocklington's theorem.