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 thateither $6k+1$ or $6k-1$ is not a prime. The results can sieve primesand 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 PrimeConjecture but easier to approach.