Combinatorial Primality Test

In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then(n - 1/m) equivalent to (-1)(n -n/2) (mod n), m = n - 1/2.This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then (n - 1/m) equivalent to 1 (mod n), m = n - 1/2, and (2) If an integer of the form n = 4k + 3, k >= 0 is prime, then (n - 1/m) equivalent to -1 (mod n), m = n - 1/2. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.