Primality proof for n = 12811352796235023217778801482819:

Take b = 2.

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

726928914639303991 is prime.
b^((n-1)/726928914639303991)-1 mod n = 6534906295293682835644564431795, which is a unit, inverse 2638829258957252504571109598233.

(726928914639303991) divides n-1.

(726928914639303991)^2 > n.

n is prime by Pocklington's theorem.