Primality proof for n = 1301:
Take b = 2.
b^(n-1) mod n = 1.
13 is prime.
b^((n-1)/13)-1 mod n = 77, which is a unit, inverse 490.
5 is prime.
b^((n-1)/5)-1 mod n = 162, which is a unit, inverse 779.
(5^2 * 13) divides n-1.
(5^2 * 13)^2 > n.
n is prime by Pocklington's theorem.