# Some remarks on the first Hardy-Littlewood conjecture

## TL;DR

This paper discusses the first Hardy-Littlewood conjecture by exploring prime distributions, providing a simplified proof of Bruns theorem, and applying numerical methods like Monte Carlo and Low discrepancy Sequences to analyze the conjecture's convergence.

## Contribution

It offers an empirical analysis of twin prime distribution, a simplified proof of Bruns theorem, and demonstrates numerical approaches to support the conjecture's validity.

## Key findings

- Twin primes distribution in classes mod(10) analyzed empirically
- Simplified proof of Bruns theorem provided
- Numerical methods support convergence of the conjecture

## Abstract

Starting from the first Hardy-Littlewood conjecture some topics will be covered: an empirical approach to the distribution of the twin primes in classes mod(10) and a simplified proof of the Bruns theorem . Finally, it will be explored an approach based on numerical analysis: Monte Carlo Method and Low discrepancy Sequences will be used to prove the convergence of the conjecture to the expected values.