A Sieve for Cousin Primes

TL;DR

This paper develops a specialized sieve to identify and analyze twin primes separated by four, providing insights into their distribution and asymptotic behavior without parity issues.

Contribution

It introduces a novel sieve method for twin primes at distance four, characterizing twin-4 ranks and non-ranks, and derives an asymptotic law for their distribution.

Findings

Constructed a sieve for twin primes at distance 4

Derived a Legendre-type sum for twin-4 ranks

Established the asymptotic distribution law

Abstract

A sieve is constructed for twin primes at distance 4, which are of the form 3(2m+1)+/-2, and are characterized by their twin-4 rank 2m+1. It has no parity problem. Non-ranks are identified as all other odd numbers and counted using odd primes p>=5. Twin-4 ranks and non-ranks make up the set of odd numbers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type sum for the number of twin-4 ranks. Due to considerable cancellations in it, the asymptotic law of its main term has the expected form and magnitude of its coefficient.