[Paper Review] Twin Prime Sieve

Hans J. Weber|arXiv (Cornell University)|Mar 23, 2012

Analytic Number Theory Research3 references3 citations

TL;DR

This paper introduces a novel sieve for twin primes of the form 6m±1 by focusing on their twin rank m, using odd primes p≥5 to identify and count non-ranks. It derives a Legendre-type formula for estimating the number of twin-ranks near primorial arguments, leveraging regularities in non-rank distribution without parity issues.

ABSTRACT

A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set of positive integers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type formula for the number of twin-ranks near primorial arguments.

Motivation & Objective

• To develop a sieve for twin primes of the form 6m±1 that avoids the parity problem common in traditional methods.
• To characterize twin primes through their twin rank m, enabling systematic analysis of their distribution.
• To identify and count non-ranks—positive integers that are not twin ranks—using odd primes p≥5.
• To exploit regularities in non-rank distribution to derive a Legendre-type formula for estimating twin-rank counts near primorial arguments.

Proposed method

• The sieve operates by characterizing twin primes via their twin rank m, treating m as the fundamental index for twin prime pairs 6m±1.
• Non-ranks are identified as integers m for which 6m±1 are not both prime, using divisibility conditions with odd primes p≥5.
• The method employs inclusion-exclusion principles over primes p≥5 to systematically eliminate non-ranks from the set of positive integers.
• Regular patterns in the distribution of non-ranks are analyzed to enable estimation of their density and cumulative count.
• The approach constructs a Legendre-type formula by adapting classical prime counting techniques to the twin-rank framework.
• The formula estimates the number of twin-ranks up to a given primorial argument, leveraging the periodic and symmetric structure of non-ranks.

Experimental results

Research questions

• RQ1 How can a sieve for twin primes be constructed without encountering the parity problem inherent in traditional methods?
• RQ2 What regularities exist in the distribution of non-ranks—integers m for which 6m±1 are not both prime—and how can they be exploited?
• RQ3 Can a Legendre-type formula be derived for estimating the number of twin-ranks near primorial arguments using non-rank regularities?
• RQ4 To what extent do the properties of odd primes p≥5 govern the structure of non-ranks in the twin-rank framework?

Key findings

• The sieve successfully avoids the parity problem by focusing on twin ranks m rather than direct prime sieving.
• Non-ranks are fully characterized by divisibility conditions involving odd primes p≥5, enabling their systematic identification and counting.
• Regular patterns in non-rank distribution allow for the derivation of a Legendre-type formula for twin-rank estimation.
• The formula provides a quantitative estimate of the number of twin-ranks near primorial arguments, offering a new analytical tool for twin prime distribution.
• The method establishes a direct link between the structure of non-ranks and the density of twin primes in the 6m±1 form.
• The approach enables a more precise and structured analysis of twin prime frequency compared to conventional sieving techniques.