Paper 2019/125

Combinatorial Primality Test

Maheswara Rao Valluri

Abstract

This paper provides proofs of the results of Laisant - Beaujeux: (1) If an integer of the form $n=4k+1$, $k>0$ is prime, then $\left(\begin{array}{c}n-1\\ m\end{array}\right)\equiv 1(modn),m=\frac{n-1}{2}$, and (2) If an integer of the form $n=4k+3$, $k\ge 0$ is prime, then $\left(\begin{array}{c}n-1\\ m\end{array}\right)\equiv -1(modn),m=\frac{n-1}{2}$. 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 $4k+3$.