A Prime Sieve Method

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,

$k \not \in (S_{l1} \cup S_{l2}) \Rightarrow 6k-1 \in \mathbb{P}$

and $k \not \in (S_{r1} \cup S_{r2}) \Rightarrow 6k+1 \in

\mathbb{P},$ where

$S_{l1}=\{k|k=(6I-1)*W+I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6IW-W+I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6xy+(x-y), x,y \in \mathbb{N}, x \leq y\}.$

$S_{l2}=\{k|k=(6I+1)*W-I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6IW+W-I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6xy-(x-y), x,y \in \mathbb{N}, x \leq y\}.$

$S_{r1}=\{k|k=(6I-1)*W-I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6IW-W-I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6xy-(x+y), x,y \in \mathbb{N}, x \leq y\}.$

$S_{r2}=\{k|k=(6I+1)*W+I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6IW+W+I, W \in \mathbb{N}, I \leq W, I \in \mathbb{N}\}\\

=\{k|k=6xy+(x+y), x,y \in \mathbb{N}, x \leq y\}.$

We also propose $6k\pm1$ Conjecture that is equivalent to Two Prime

Conjecture but easier to approach.