Number Theory pp 363-398

# The Number of Prime Numbers

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

## Abstract

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 $$\propto_1, \ldots, \propto_n$$.

## Keywords

Prime Number Zeta Function Prime Divisor Algebraic Number Simple Pole
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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