Primality proof for n = 20705423504133292078628634597817:

Take b = 2.

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

862725979338887169942859774909 is prime.
b^((n-1)/862725979338887169942859774909)-1 mod n = 16777215, which is a unit, inverse 9528120837126448207374882683518.

(862725979338887169942859774909) divides n-1.

(862725979338887169942859774909)^2 > n.

n is prime by Pocklington's theorem.