Primality proof for n = 2953:
Take b = 3.
b^(n-1) mod n = 1.
41 is prime.
b^((n-1)/41)-1 mod n = 36, which is a unit, inverse 2871.
3 is prime.
b^((n-1)/3)-1 mod n = 799, which is a unit, inverse 717.
(3^2 * 41) divides n-1.
(3^2 * 41)^2 > n.
n is prime by Pocklington's theorem.