6m Theorem for Prime numbers

TL;DR

This paper presents a new theorem identifying conditions under which numbers of the form 6^{m+1}·N - 1 are prime, extending known prime patterns for specific N values and discussing criteria for larger N.

Contribution

The paper introduces a novel theorem characterizing prime numbers of a specific form involving powers of 6 and parameter N, with conditions for N greater than 13.

Findings

Identifies prime numbers of the form 6^{m+1}·N - 1 for certain N values.

Provides conditions involving modular arithmetic for N > 13.

Extends prime pattern understanding for numbers of this form.

Abstract

We show that for any $P= 6^{m+1}.N -1 $ is a prime number for any $1 < N \le 13$ , $N \ne 8$ and $N \ne i^{m+1}Mod(6i+1) $ where $ i \in Z^+ $ and $ m \in $ $odd$ $Z^+ $ for $1 < N \le 13$ and $N \ne 8$ and also we further discussed that $P= 6^{m+1}.N -1 $ is a prime number for $ N >13 $ if and only if , $N \ne i^{m+1}Mod(6i+1) +(6i +1)a $ $ ; i,a \le Z^+ $