Towards Proving the Twin Prime Conjecture using a Novel Method For Finding the Next Prime Number ${P_{N+1}}$ after a Given Prime Number ${P_{N}}$ and a Refinement on the Maximal Bounded Prime Gap ${G_{i}}$

TL;DR

This paper proposes a new method to identify the next prime after a given prime and introduces a system of inequalities that, if proven to have infinite solutions, could prove the Twin Prime Conjecture, along with a new upper bound on prime gaps.

Contribution

It introduces a novel approach to find the next prime and formulates inequalities linked to twin primes, offering a potential pathway to prove the Twin Prime Conjecture.

Findings

Derived a system of inequalities related to twin primes

Proposed a new upper bound on prime gaps

Established a connection between solutions to inequalities and the Twin Prime Conjecture

Abstract

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is a twin prime. Twin primes are primes having a prime gap of ${2}$. The pairs ${(5,7),(11,13),(41,43)}$, etcetera are all twin primes. This paper envisions that if the proposed system of inequalities can be proven to have infinite solutions, the Twin Prime Conjecture will evidently be proven true. The paper also derives a novel upper bound on the prime gap, ${G_{i}}$ between ${P_{i} \; and \; P_{i+1}}$, as a function of ${P_{i}}$.