A probabilistic approach to the twin prime and cousin prime conjectures

TL;DR

This paper uses a probabilistic method to analyze the distribution of twin and cousin primes within specific intervals, proposing a new conjecture on their infinitude and showing they share similar asymptotic behaviors.

Contribution

It introduces a probabilistic framework partitioning integers into intervals based on consecutive primes and demonstrates that the probability of containing twin or cousin primes approaches one, leading to a new conjecture.

Findings

Probability of intervals containing twin or cousin primes approaches 1 as n increases

Proposes a new conjecture on the infinitude of twin and cousin primes

Shows twin and cousin primes have similar asymptotic distributions

Abstract

We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$ and $p_n$ being two consecutive primes, and evaluates the probability $q_n$ that such an interval contains a twin prime and a cousin prime. Combining Merten's third theorem with the properties of the binomial distribution, we show that $q_n$ approaches $1$ as $n \to \infty$. A study of the convergence properties of the sequence $\{q_n\}$ allows us to propose a new, more stringent conjecture concerning the existence of infinitely many twin and cousin primes. In accord with the Hardy-Littlewood conjecture, it is also shown that twin and cousin primes share the same asymptotic distribution.