Numerical analysis approach to twin primes conjecture

Symmetry

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undeleted (that is, 1) integers in a modified version of the Sieve procedure, in such a way a symmetric succession of runs of zeroes is found in correspondence of the gaps between the undeleted integers in each period. Such a formulation is able to estimate the prime number function in an equivalent way to the logarithmic integral function Li(x). The present analysis is then extended to the twin prime...

This paper brings up a few possible approaches to solving the twin primes conjecture. [Published in international mathematics journal. Acknowledgments: The author is thankful to the referees and to the Editor-in-Chief for their insightful comments which led to an improvement in this paper.]

2021

We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter ${\lambda}_x$. An estimation of the number of twin primes obtained is called the fundamental structure of the distribution of twin primes. Combining the latter with an asymptotic bound of ${\lambda}_x$ establishes venues conducive to an asymptotic bound of the number of twin primes less than $x$.

Lettera Matematica, 2017

2019

While the notion of prime numbers has existed for millennia, twin primes have only been around for little over a century. Although it is not known whether there are infinitely many twin primes, the prime gap was very recently shown to be no greater than 246. The fact that the summed reciprocals of twin primes converge to approximately 1.9 has also been demonstrated. It has further been established that there do exist infinitely many primes p for which p+2 is the product of no more than two primes. A criterion for twin primes does exist but it is neither sufficient to show the existence of an infinite number of them, nor feasible as a computational tool.

Lambert Publishing Academy , 2017

AIP Conference Proceedings, 2018

viXra, 2012

The author had published a paper on the solutions for the twin primes conjecture in an international mathematics journal in 2003. This paper, which consists of 2 parts that are each self-contained, presents some approaches to the twin primes problem.

Academia Letters, 2021

ZPE 223, 2022, 219ff.