On the prime distribution

TL;DR

This paper claims to establish formulas and sequences related to prime distribution, providing proofs for the twin prime and Goldbach conjectures, which are longstanding open problems in number theory.

Contribution

It introduces a new estimation formula for prime counts and constructs sequences to prove twin prime and Goldbach conjectures.

Findings

Proof of the twin prime conjecture.

Proof of the Goldbach conjecture for even integers greater than 2700.

Construction of sequences with increasing prime pairs.

Abstract

In this paper, the estimation formula of the number of primes in a given interval is obtained by using the prime distribution property. For any prime pairs $p>5$ and $ q>5 $, construct a disjoint infinite set sequence $A_1, A_2, \ldots, A_i. \ldots $, such that the number of prime pairs ($p_i$ and $q_i $, $p_i-q_i = p-q $) in $A_i $ increases gradually, where $i>0$. So twin prime conjecture is true. We also prove that for any even integer $m>2700$, there exist more than 10 prime pairs $(p,q)$, such that $p+q=m$. Thus Goldbach conjecture is true.