Prime

All primes can be indexed by $k$, as primes must be in the form of

$6k+1$ or $6k-1$. In this paper, we explore for what $k$ such that

either $6k+1$ or $6k-1$ is not a prime. The results can sieve primes

and especially twin primes.

$k \in S_{l} \Rightarrow 6k-1 \not \in \mathbb{P}$, $k \in S_{r}

\Rightarrow 6k+1 \not \in \mathbb{P},$ where $S_{l} = [-I]_{6I+1} =

[I]_{6I-1} \backslash \min([I]_{6I-1}), I \in \mathbb{N},$ and

$S_{r} = [-I]_{6I-1} \cup [I]_{6I+1} \backslash \min([I]_{6I+1}), I

\in \mathbb{N}.$ That is,