A recursion equation for prime numbers

TL;DR

This paper introduces a recursion equation that determines the next prime number based on the previous primes, using a limit involving infinite series and products, providing a novel analytical approach to prime number generation.

Contribution

It presents a new recursion formula for prime numbers derived from infinite series and products, linking prime generation to analytical limits.

Findings

The recursion formula accurately predicts the next prime from previous primes.

The sum limit can be replaced by a finite bound without losing validity.

The approach offers a new perspective on prime number distribution.

Abstract

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}. $$ The upper limit on the sum can be replaced by $2p_n -1$, and the result still holds.