Number of Primes is Infinite/Proof 4

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Theorem

The number of primes is infinite.


Proof

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

From Euler Product for Riemann Zeta Function:

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

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 Sum of Reciprocals 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.


Sources