Primality proof for n = 39863:

Take b = 2.

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

1049 is prime.
b^((n-1)/1049)-1 mod n = 39211, which is a unit, inverse 2629.

(1049) divides n-1.

(1049)^2 > n.

n is prime by Pocklington's theorem.