Primality proof for n = 10303:
Take b = 2.
b^(n-1) mod n = 1.
101 is prime.
b^((n-1)/101)-1 mod n = 8577, which is a unit, inverse 4471.
17 is prime.
b^((n-1)/17)-1 mod n = 4533, which is a unit, inverse 9012.
(17 * 101) divides n-1.
(17 * 101)^2 > n.
n is prime by Pocklington's theorem.