A recursion equation for prime numbers

Abstract

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

2 pages; replacement 10/05/2008 corrects typographical error page 2, reference #1, author last name