Many guides will refer to Euler's product formula as simple way to prove that the number of primes is infinite.
$$\sum_n\frac{1}{n} = \prod_p \frac{1}{1-\frac{1}{p}}$$
The argument is that if the primes were finite, the product on the right hand side is finite, noting that $1-\frac{1}{p}$ is never zero.
However, the product formula itself is constructed by application of the fundamental theorem of arithmetic to an infinite series with terms involving only primes.
Does this mean such proofs are a circular argument – because they use a product formula whose construction depends entirely on the infinitude of primes?
Best Answer
To prove the equality you need that every natural number is uniquely represented as a product of primes. It does not need the fact that the set of primes is infinite. In fact to prove that the set of primes is infinite, you do not need Euler equality. You only need inequality $LHS \le RHS$ which follows from the fact that every number is a product of primes (uniqueness is not needed). If you compare that proof with Euclid's original proof, it is not clear which one uses less prior infomation about primes.