Is this a novel factorization theorem? Solving $\sin^2(\pi x)+\sin^2\left(\frac{\pi p}{x}\right)=0$ for $x$ gives the integer factors of $p$.

Question

I have found that $$ \sin^2(\pi x) + \sin^2\left(\frac{\pi p}{x}\right)=0 $$

solved for $x$ defines the integer factors of $p$. Iteratively applied it also defines the prime numbers smaller than $n$ where $p=n!$.

Why isn't this mentioned anywhere? Surely this is right up there with (although not related to) the Weierstrass factorization theorem? Has no one noticed that before, or is it too trivial to mention?

I've added an interactive demonstration at: https://www.desmos.com/calculator/ls1kp8i78n

Answer 1

This formula immediately translates to

$$x\in\mathbb Z\land x|p.$$

Nothing really special.