A Prime Sieve Method

A Prime Sieve Method

Citation Author(s):
Wei
Ren
Submitted by:
Wei Ren
Last updated:
Wed, 01/30/2019 - 03:47
DOI:
10.21227/8j5c-4m32
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Abstract: 

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.

Instructions: 

ANSI C source code can be complied by any C complier. 

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[1] Wei Ren, "A Prime Sieve Method ", IEEE Dataport, 2019. [Online]. Available: http://dx.doi.org/10.21227/8j5c-4m32. Accessed: May. 28, 2020.
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doi = {10.21227/8j5c-4m32},
url = {http://dx.doi.org/10.21227/8j5c-4m32},
author = {Wei Ren },
publisher = {IEEE Dataport},
title = {A Prime Sieve Method },
year = {2019} }
TY - DATA
T1 - A Prime Sieve Method
AU - Wei Ren
PY - 2019
PB - IEEE Dataport
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Wei Ren. (2019). A Prime Sieve Method . IEEE Dataport. http://dx.doi.org/10.21227/8j5c-4m32
Wei Ren, 2019. A Prime Sieve Method . Available at: http://dx.doi.org/10.21227/8j5c-4m32.
Wei Ren. (2019). "A Prime Sieve Method ." Web.
1. Wei Ren. A Prime Sieve Method [Internet]. IEEE Dataport; 2019. Available from : http://dx.doi.org/10.21227/8j5c-4m32
Wei Ren. "A Prime Sieve Method ." doi: 10.21227/8j5c-4m32