Primality proof for n = 63863:

Take b = 2.

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

863 is prime.
b^((n-1)/863)-1 mod n = 16462, which is a unit, inverse 28859.

(863) divides n-1.

(863)^2 > n.

n is prime by Pocklington's theorem.