Proof of Infinitude of Primes by Euler's Product Formula is Circular? 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?

No, the proof is not circular. If we assume there are a finite number of primes $p_1,\ldots,p_k$, then we would assume that any $n\in\mathbb{N}$ would be able to be factorized as $p_1^{\alpha_1}\ldots p_k^{\alpha_k}$. (The proof of the Fundamental Theorem of Arithmetic does not require that there be an infinite number of primes.) This would lead to the Euler product formula, which we would then use to provide the contradiction, once we have shown that $\sum_{n\in\mathbb{N}} \frac1n$ is infinite.
