Primality proof for n = 1009:

Take b = 2.

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

7 is prime.
b^((n-1)/7)-1 mod n = 104, which is a unit, inverse 456.

3 is prime.
b^((n-1)/3)-1 mod n = 373, which is a unit, inverse 211.

(3^2 * 7) divides n-1.

(3^2 * 7)^2 > n.

n is prime by Pocklington's theorem.