Elementary Primes Counting Methods

TL;DR

This paper introduces elementary sieve-based proofs for counting specific prime subsets, such as twin primes, Germain primes, and quadratic primes, aiming to support their conjectured infinitude.

Contribution

It provides new elementary proof techniques for prime counting problems related to twin, Germain, and quadratic primes, using a weighted sieve approach.

Findings

Elementary proofs for twin primes, Germain primes, and quadratic primes.

Supports the conjecture that these prime subsets are infinite.

Introduces a novel weighted sieve method for prime counting.

Abstract

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the subset of Germain primes PG = {p and 2p + 1 are primes}, and the subset of quadratic primes Pf = {p = n^2 + 1 primes}. These subsets of primes are widely believed to be infinite subsets of prime numbers.