Primality proof for n = 352576608953991537322303:
Take b = 2.
b^(n-1) mod n = 1.
16446708426521 is prime.
b^((n-1)/16446708426521)-1 mod n = 250478389233923669204206, which is a unit, inverse 220586105970312567109739.
(16446708426521) divides n-1.
(16446708426521)^2 > n.
n is prime by Pocklington's theorem.