Euclid had already proven, around 300 BC, that there are infinitely many prime numbers. In 1737 Euler found a completely different proof, which linked the primes to calculus.

For each prime $p$, the sum of a geometric progression gives

$$\frac{1}{1 - \frac{1}{p}} = 1 + \frac{1}{p} + \frac{1}{p^2} + \frac{1}{p^3} + \cdots$$

Multiply these expressions to $p = 2, 3, 5, 7, \dots$ Each product of terms has the form $\frac{1}{2^a 3^b 5^c \cdots}$. As every natural number is written in a single way as a product of primes, each fraction $\frac{1}{n}$ appears exactly once:

$$\prod_{p \text{ primo}} \frac{1}{1 - \frac{1}{p}} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots$$

But the sum on the right, the harmonic series, grows without limit: the terms from $\frac{1}{3}$ to $\frac{1}{4}$ add up to more than $\frac{1}{2}$, those from $\frac{1}{5}$ to $\frac{1}{8}$ too, those from $\frac{1}{9}$ to $\frac{1}{16}$ as well, and so on. If there were a finite amount of primes, the left side would be a product of finite numbers, therefore a finite number. Contradiction: there are infinitely many primes.

Grouping the terms into blocks of 1, 2, 4, 8… each block goes from 1/2. Adding infinite blocks, the sum keeps growing.

Euler went further and showed that even the sum $\frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \cdots$ of the inverses of the primes grows without limit. This bridge between primes and analysis gave rise to thetheory number analytics.

All topics by Leonhard Euler · ← $V - A + F = 2$ and the soccer ball · The Fermat number that was not prime →

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