Primality proof for n = 7071763:
Take b = 2.
b^(n-1) mod n = 1.
89 is prime.
b^((n-1)/89)-1 mod n = 1152977, which is a unit, inverse 2200779.
41 is prime.
b^((n-1)/41)-1 mod n = 2631218, which is a unit, inverse 3872863.
(41 * 89) divides n-1.
(41 * 89)^2 > n.
n is prime by Pocklington's theorem.