6m Theorem for Prime numbers

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 , $N \ne i^{m+1}Mod(6i+1) +(6i +1)a$ $; i,a \le Z^+$