Joseph B. Keller
Published 2007-11-26, updated 2008-10-05Version 2
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.
Comments: 2 pages; replacement 10/05/2008 corrects typographical error page 2, reference #1, author last name
Improving the Estimates for a Sequence Involving Prime Numbers
On a sequence involving the prime numbers
Some properties of a sequence defined with the aid of prime numbers
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