A new enlightenment about gaps between primes

TL;DR

This paper introduces a novel approach using co-prime sequences to prove longstanding conjectures in number theory, including the infinitude of twin primes and Goldbach's conjecture, by analyzing properties of co-prime sets and their linear combinations.

Contribution

The paper presents a new method based on co-prime sets to prove multiple major conjectures in prime number theory, including twin primes and Goldbach's conjecture.

Findings

Proves infinitely many twin primes.

Establishes Goldbach conjecture.

Derives Polignac's conjecture and even number representations.

Abstract

The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea of our approach is summarized on the creation and on the analyzing sequence of sets of distinct co-primes with the first $n$ primes, $\left\{ p_i :\, i\leq n \right\}$, and the important properties of the modulus linear combination of the co-prime sets, $H=\left(1,p_{n+1},..., \Pi_{i=1}^n p_i-1\right) $, that gives sets of even numbers $\{0,2,4,..., \Pi_{i=1}^n p_i -2 \}$. Furthermore, by generalizing our approach, the Polignac conjecture "the existence of infinitely many cousin primes, $p_{n+1}-p_{n}=4$," and the statement that "every even integer can be expressed as a difference of two primes," are derived as well.