  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.
