A probabilistic approach to the twin prime and cousin prime conjectures

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 $n\to \mathrm{\infty }$. 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.