Primality proof for n = 1783:
Take b = 2.
b^(n-1) mod n = 1.
11 is prime.
b^((n-1)/11)-1 mod n = 366, which is a unit, inverse 1252.
3 is prime.
b^((n-1)/3)-1 mod n = 192, which is a unit, inverse 1718.
(3^4 * 11) divides n-1.
(3^4 * 11)^2 > n.
n is prime by Pocklington's theorem.