Primality proof for n = 353:
Take b = 3.
b^(n-1) mod n = 1.
11 is prime.
b^((n-1)/11)-1 mod n = 139, which is a unit, inverse 160.
2 is prime.
b^((n-1)/2)-1 mod n = 351, which is a unit, inverse 176.
(2^5 * 11) divides n-1.
(2^5 * 11)^2 > n.
n is prime by Pocklington's theorem.