Number of Primes is Infinite/Proof 4

Theorem

The number of primes is infinite.

Proof

Aiming for a contradiction, suppose there exist only a finite number of primes.

$\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$

When $z = 1$ this gives:

$\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 n = \prod_p \frac 1 {1 - 1/p}$

As by hypothesis there exist only a finite number of primes, the right hand side is also finite.

But from Harmonic Series is Divergent, the left hand side diverges to infinity.

The result follows by Proof by Contradiction.

$\blacksquare$

Historical Note

This proof that the number of primes is infinite was devised by Leonhard Paul Euler.