Primality proof for n = 1300347341:

Take b = 2.

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

3824551 is prime.
b^((n-1)/3824551)-1 mod n = 1098383323, which is a unit, inverse 1281468091.

(3824551) divides n-1.

(3824551)^2 > n.

n is prime by Pocklington's theorem.