Primality proof for n = 6500537:
Take b = 2.
b^(n-1) mod n = 1.
103 is prime.
b^((n-1)/103)-1 mod n = 5010448, which is a unit, inverse 270428.
23 is prime.
b^((n-1)/23)-1 mod n = 4901119, which is a unit, inverse 803080.
7 is prime.
b^((n-1)/7)-1 mod n = 2957565, which is a unit, inverse 1102302.
(7^3 * 23 * 103) divides n-1.
(7^3 * 23 * 103)^2 > n.
n is prime by Pocklington's theorem.