Primality proof for n = 560513308871:

Take b = 2.

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

411679 is prime.
b^((n-1)/411679)-1 mod n = 446426860155, which is a unit, inverse 21549872957.

8009 is prime.
b^((n-1)/8009)-1 mod n = 515895509753, which is a unit, inverse 317410781157.

(8009 * 411679) divides n-1.

(8009 * 411679)^2 > n.

n is prime by Pocklington's theorem.