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.