I know there are many questions on the site about finding a proof that π is irrational, but I'm posting the question separately to discuss a particular proof further
We know that the Wallis Product is :
$$\frac{π}{2}=(\frac{2}{1}\cdot\frac{2}{3})(\frac{4}{3}\cdot\frac{4}{5})(\frac{6}{5}\cdot\frac{6}{7})(\frac{8}{7}\cdot\frac{8}{9})\cdots$$
This means that if $\pi$ is a rational number, its numerator will be an even number and its denominator will be an odd number
After that, all we have to do is find a formula for the number $\pi$ that gives a "reversed" fraction whose numerator is an odd number and whose denominator is an even number. Thus, we obtain a proof similar to the classical proof that $\sqrt{2}$ is irrational. Indeed, after some research, I found formula of this model that are attributed to Leonard Euler:
Assuming that $p_n$ is a notation that refers to the prime number $n$, the formula we want is :
$$\frac{π}{4}=\prod_{n=1}^∞ (\frac{p_n}{p_n+(-1)^{\frac{p_n+1}{2}}})=\frac{3}{3+1}\cdot\frac{5}{5-1}\cdot\frac{7}{7+1}\cdot\frac{11}{11+1}\cdot\frac{13}{13-1}\cdots$$
It represents an odd number divided by an even number as required. Thus, we obtain a contradiction showing that $\pi$ is irrational.
My question is : Is my proof valid or is there an error in this proof?
I don't know if I can deduce this from an infinite ratio or if it is invalid. If this is not true please give an example of a rational number that has two representations as an infinite product of the two opposite forms.