Primality proof for n = 294639853:

Take b = 2.

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

461 is prime.
b^((n-1)/461)-1 mod n = 204473882, which is a unit, inverse 57716566.

241 is prime.
b^((n-1)/241)-1 mod n = 198273527, which is a unit, inverse 100964630.

(241 * 461) divides n-1.

(241 * 461)^2 > n.

n is prime by Pocklington's theorem.