Euler's Proof

From Euclid to Euler

Euclid proved that there are infinitely many primes by contradiction. Euler discovered a very different proof based on infinite series and products.

His argument connects prime numbers with the harmonic series

\sum_{n=1}^{\infty}\frac1n.

This series diverges:

1+\frac12+\frac13+\frac14+\cdots=\infty.

Euler observed that the harmonic series can be decomposed multiplicatively using prime factorization. This insight became one of the foundational ideas of analytic number theory.

The Harmonic Series

Consider the finite partial sums

1+\frac12+\frac13+\cdots+\frac1N.

As $N\to\infty$ , these sums grow without bound.

The divergence of the harmonic series is classical. One proof groups terms:

1+\frac12 + \left(\frac13+\frac14\right) + \left(\frac15+\cdots+\frac18\right) +\cdots

Each block after the first has sum at least

\frac12.

Since infinitely many such blocks occur, the total sum diverges.

Euler’s idea was to express this same series using primes.

Euler Product Formula

For a prime $p$ , consider the geometric series

1+\frac1p+\frac1{p^2}+\frac1{p^3}+\cdots.

Since

\left|\frac1p\right|<1,

the geometric series formula gives

1+\frac1p+\frac1{p^2}+\cdots = \frac1{1-\frac1p}.

Now multiply these series over all primes:

\prod_p \left( 1+\frac1p+\frac1{p^2}+\cdots \right).

Expanding the product formally produces terms of the form

\frac1{p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}}.

By unique prime factorization, every positive integer appears exactly once in this expansion. Therefore

\prod_p \left( 1+\frac1p+\frac1{p^2}+\cdots \right) = \sum_{n=1}^{\infty}\frac1n.

Using the geometric-series identity,

\sum_{n=1}^{\infty}\frac1n = \prod_p \frac1{1-\frac1p}.

This is Euler’s product formula for the harmonic series.

The Contradiction

Suppose there were only finitely many primes:

p_1,p_2,\ldots,p_k.

Then Euler’s product formula would become

\sum_{n=1}^{\infty}\frac1n = \prod_{i=1}^{k} \frac1{1-\frac1{p_i}}.

The right-hand side is a finite product of finite numbers, so it is finite.

But the left-hand side is the harmonic series, which diverges.

This contradiction shows that there cannot be finitely many primes.

Therefore infinitely many primes exist.

Why the Product Works

The key idea is unique prime factorization.

When the infinite product is expanded, each integer appears exactly once because every integer has exactly one prime decomposition.

For example,

\frac1{12} = \frac1{2^2\cdot3}

arises uniquely from choosing

\frac1{2^2}

from the factor corresponding to $2$ , choosing

\frac13

from the factor corresponding to $3$ , and choosing $1$ from all remaining prime factors.

No other choice produces $1/12$ .

Thus prime factorization converts an additive object, the harmonic series, into a multiplicative product over primes.

A More Refined View

Euler’s proof shows more than infinitude. It suggests that primes govern the structure of integers multiplicatively.

The identity

\sum_{n=1}^{\infty}\frac1n = \prod_p \frac1{1-\frac1p}

is the first example of an Euler product.

Later, Euler generalized this to

\sum_{n=1}^{\infty}\frac1{n^s} = \prod_p \frac1{1-p^{-s}}, \qquad s>1.

The left-hand side is now the Riemann zeta function:

\zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}.

This identity became one of the central formulas of analytic number theory.

Divergence of the Prime Reciprocal Sum

Euler proved an even stronger statement:

\sum_p\frac1p

diverges.

Thus not only are there infinitely many primes, but their reciprocals still accumulate enough mass to produce divergence.

This result shows that primes are not excessively sparse.

The proof uses logarithms applied to Euler products:

\log\prod_p\frac1{1-p^{-1}} = -\sum_p\log\left(1-\frac1p\right).

Using the approximation

-\log(1-x)\approx x

for small $x$ , one obtains behavior comparable to

\sum_p\frac1p.

Since the harmonic series diverges, the reciprocal prime sum must also diverge.

Euclid Versus Euler

Euclid’s proof is combinatorial and constructive. It creates a number not divisible by a given finite set of primes.

Euler’s proof is analytic. It studies infinite sums and products and uses divergence.

The two proofs illustrate two major directions in number theory:

divisibility and arithmetic structure,
analytic behavior and asymptotic growth.

Modern number theory combines both viewpoints extensively.

Role in Number Theory

Euler’s proof marks the beginning of analytic number theory. It reveals that primes are connected not only with divisibility, but also with infinite series, products, and analytic behavior.

The idea that arithmetic information can be encoded in analytic functions eventually leads to the zeta function, Dirichlet $L$ -functions, the prime number theorem, and the Riemann hypothesis.

Thus Euler’s proof is not merely another proof of the infinitude of primes. It is the beginning of an entirely new way to study arithmetic.