Primality proof for n = 75445702479781427272750846543864801:

Take b = 2.

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

1919519569386763 is prime.
b^((n-1)/1919519569386763)-1 mod n = 56747016255153332959347766392083458, which is a unit, inverse 74602835786463724888147869570305675.

72106336199 is prime.
b^((n-1)/72106336199)-1 mod n = 71619851993489548214160924360205393, which is a unit, inverse 55965016733432503242248684386325671.

(72106336199 * 1919519569386763) divides n-1.

(72106336199 * 1919519569386763)^2 > n.

n is prime by Pocklington's theorem.