I was wondering if I could receive assistance for the following system:
$$\begin{cases}(x/a)^{3.2}+(y/b)^{3.2}=1\\ a/b = 174.1/86\end{cases}$$ I'm looking for integer solutions or how to find them (if possible).
Edit: I have currently gotten this far:
$$\begin{cases} ((86x)^{3.2}+y^{3.2})/(174.1b)^{3.2}=1 \end{cases}$$
Thanks in advance!
Ken
You don't need the equation for $a/b$. There is no solution of the first equation in integers $a,b,x,y$ with $x, y \ne 0$.
You want $r = x/a$ and $s = y/b$ to be nonzero rational numbers, with $r^{16/5} + s^{16/5} = 1$. Let $r^{1/5} = t$ and $s^{1/5} = u$, so that $t^{16} + s^{16} = 1$. By a case of Fermat's Last Theorem, it is known that this has no rational solutions with $s,t\ne 0$. So at least one of $t$ and $u$ is irrational. WLOG assume $t$ is irrational, i.e. $r$ is not the $5$'th power of a rational number. Now the same is true of $r^{16}$. It can be shown for any rational $y$ that is not the $5$'th power of a rational, the polynomial $X^5 - y$ is irreducible. So $u = t^{16}$, a root of the polynomial $X^5 - r^{16}$, has the minimal polynomial $X^5 - r^{16}$.
Now $s^{16} = (1 - r^{16/5})^5 = (1 - t^{16})^5 = (1-u)^5$, and thus $(X-1)^5 + s^{16}$ is a different monic polynomial of degree $5$ with $u$ as a root. That contradicts the assertion that $X^5 - r^{16}$ is the minimal polynomial of $u$.
We conclude that no such solution is possible.