Primality proof for n = 7447165497053341:

Take b = 2.

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

716223737 is prime.
b^((n-1)/716223737)-1 mod n = 2172906263134938, which is a unit, inverse 1988971935379545.

(716223737) divides n-1.

(716223737)^2 > n.

n is prime by Pocklington's theorem.