Euler's proof of the infinitude of primes

Almost all of us are familiar with Euclid's proof regarding the infinitude of primes but in this post I will discuss the way Leonhard Euler approached it.I didn't find this proof on anywhere on Brilliant so I decided to do a post on this.

Leonhard Euler Image Credit:Wikipedia

$\textbf{Proof:}$According to the fundamental theorem of Arithmetic every integer$n$ can be represented as the product of prime numbers $n = p_1^{a_1} p_2^{a_2}...... p_k{a_k}$.

Let us assume there are finitely many primes $p_1<p_2<p_3<......<p_n$.

Let us denote by $N$ the following product:

$N = (1 + \frac{1}{p_1}+\frac{1}{p_1^{2}}+\frac{1}{p_1^{3}}+...)(1 + \frac{1}{p_2}+\frac{1}{p_2^{2}}+\frac{1}{p_2^{3}}+..)....$ $....(1 + \frac{1}{p_n}+\frac{1}{p_n^{2}}+\frac{1}{p_n^{3}}+...)$

Now for each term in the product we can evaluate the sum of the expression using infinite GP(since $\frac{1}{p_i}<1$ ) to be : $(1 + \frac{1}{p_i}+\frac{1}{p_i^{2}}+\frac{1}{p_i^{3}}+...) = \frac{p_i}{p_i-1}$

So clearly the expression on the right hand side is finite...so N corresponds to a finite expression. - (1)

But now if we multiply out the terms in n then we will get the sum of the reciprocals of all the natural numbers because every natural number is the product of primes and each distinct combination of primes lead to a natural number.Clearly we can see that the product of the expressions in N leads to the occurrence of every possible combination of the prime numbers in the denominator and hence the reciprocals of all natural numbers . That is

$N = (1 + \frac{1}{p_1}+\frac{1}{p_1^{2}}+\frac{1}{p_1^{3}}+...)(1 + \frac{1}{p_2}+\frac{1}{p_2^{2}}+\frac{1}{p_2^{3}}+..)....$ $....(1 + \frac{1}{p_n}+\frac{1}{p_n^{2}}+\frac{1}{p_n^{3}}+...)$

$N = 1+\frac{1}{2}+\frac{1}{3}+........$

We know that this series is diverging which contracts equation 1 which proved the finiteness of N.Hence we have a contradiction!!

So there are infinitely many primes.

$\textbf{Personal Note:}$It will be extremely helpful for me if Mr.Calvin Lin and other staff members offer a bit of constructive criticism of my posts and offer some advice on how I can improve my writing skills.My peers are also welcome to do the same..Here are my previous posts:

Partitioning of integers

Fermat's Method of infinite descent

Stacking problems