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arXiv · 0711.3940

A recursion equation for prime numbers

Joseph B. Keller
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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.

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Joseph B. Keller. 2008-10-05. A recursion equation for prime numbers. https://arxiv.org/abs/0711.3940

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