On the distribution of consecutive composite odd numbers and twin primes

TL;DR

This paper investigates the distribution of twin primes and consecutive odd composite numbers, using counting arguments and congruence analysis to understand their frequency and properties.

Contribution

It introduces a new counting approach to analyze twin primes and derives possible congruences, providing insights into their distribution.

Findings

The limit of an alternating sum related to twin primes equals 1.

Twin primes are relatively rare, as indicated by the counting argument.

Possible congruences for twin primes are identified.

Abstract

We study pairs of consecutive odd numbers through a straightforward indexing. We focus in particular on twin primes and their distribution. With a counting argument, we calculate the limit of an alternating sum that is equal to 1 which means there are few twin primes. Finally, we show how to find the possible congruences for these prime numbers.