The importance of finding the upper bounds for prime gaps in order to solve the twin primes conjecture and the Goldbach conjecture

TL;DR

This paper emphasizes the significance of establishing upper bounds for prime gaps to address the twin primes and Goldbach conjectures, proposing a novel approach analyzing prime distribution using relative numbers and arrangements.

Contribution

It introduces a new method for determining upper bounds for prime gaps based on prime distribution analysis with relative numbers, differing from traditional approaches.

Findings

Proposes a unique arrangement of numbers to minimize prime gap distances.

Suggests that increasing prime powers within ranges could reduce overlaps.

Indicates potential implications for resolving major conjectures in number theory.

Abstract

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative numbers. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only arrangement that minimizes the distance between two powers having the same absolute value of the base D, with |D|<=P. This arrangement is considered important because by increasing the number of powers of the prime numbers within a range of consecutive numbers, it is presumed to decrease the overlap between the prime numbers considered. Consequently, by reducing these overlaps, we suppose to obtain an arrangement, in which the prime numbers less than and equal to P and their multiples occupy the greatest possible number of positions within a range of 2P+1 consecutive numbers. If this result could be demonstrated, would imply not only the resolution of the Legendre conjecture, but also a step forward in the resolution of the twin primes conjecture and the Goldbach conjecture.