Number Theory pp 363-398 | Cite as

The Number of Prime Numbers

  • W. A. Coppel
Part of the Universitext book series (UTX)


It was already shown in Euclid’s Elements (Book IX, Proposition 20) that there are infinitely many prime numbers. The proof is a model of simplicity: let \(p_1, \ldots, p_n\) be any finite set of primes and consider the integer \(N = p_1 \ldots p_n + 1\). Then \(N > 1\) and each prime divisor p of N is distinct from \(p_1, \ldots, p_n\), since \(p = p_j\) would imply that p divides \(N - p_1 \cdots p_n = 1\). It is worth noting that the same argument applies if we take \(N = p^{\propto_1}_1 \cdots p^{\propto_n}_n + 1\), with any positive integers \(\propto_1, \ldots, \propto_n\).


Prime Number Zeta Function Prime Divisor Algebraic Number Simple Pole 
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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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