TITLE: Proof of Infinitude of Primes by Euler's Product Formula is Circular?
QUESTION [15 upvotes]: 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?

REPLY [7 votes]: As others already noted, it's non-circular because the FTA doesn't assume infinitely many primes. In fact, the original proof there are infinitely many primes uses a fragment of the FTA. We multiply together finitely many primes and add $1$, and to continue the argument we need to know the result will have some prime factor. We prove this helpful result early in the proof of the FTA.