Given an odd integer number n, and x is an unknown odd integer number and $ 1 < x \leq n $
Can i solve the following system of equations to find $x$? If i can, how to solve it?:
$$\begin{cases} y = {\Large \frac{-\cos(\frac{\pi}{2}x)}{\log(x)}} \\ y = -\cos({\large \frac{\pi}{2}\frac{n}{x}}) \\ y = 0 \end{cases}$$
i only need 1 value for $x$, the lowest one.
You can use $n = 15$, for example. Solving this system of equations is very useful for number theory. And if it's easy, it's better. You can see it on http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiItY29zKHBpLzIqeCkvbG9nKHgpIiwiY29sb3IiOiIjRkYwMDAwIn0seyJ0eXBlIjowLCJlcSI6Ii1jb3MocGkvMioxNS94KSIsImNvbG9yIjoiIzAwRkYwMCJ9LHsidHlwZSI6MCwiZXEiOiIwIiwiY29sb3IiOiIjMDAwMEZGIn0seyJ0eXBlIjoxMDAwfV0-
Solving this problem, is part of an algorithm.
I could not go further.
If $y=0$ then you have to solve the equation $$0=-\cos\left(\frac{n\pi}{2x}\right)$$ so you will get $$\frac{n \pi}{2x}=\pm\frac{\pi}{2}+2k\pi$$