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.