Primality proof for n = 4473682817603206637471:

Take b = 2.

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

447368281760320663747 is prime.
b^((n-1)/447368281760320663747)-1 mod n = 1023, which is a unit, inverse 3769613478762428271261.

(447368281760320663747) divides n-1.

(447368281760320663747)^2 > n.

n is prime by Pocklington's theorem.