Sum Of Squares And Multiple Of Product
Given that $x$, $y$ are positive integers and $x \ne y$, consider the following two results:
$5^2 + 13^2 = 194, 5 \times 13 \times 3 = 195$
$11^2 + 41^2 = 1802, 11 \times 41 \times 4 = 1804$
We can see that in both cases that the sum of the squares of $x$ and $y$ are almost a multiple of their product.
Prove that $x^2 + y^2$ can never be multiple of $xy$.
We are considering solutions to the equation $x^2 + y^2 = kxy$, where $k$ is some positive integer.
Suppose that $GCD(x, y) = h$; let $x = hm$ and $y = hn$.
Therefore $h^2 m^2 + h^2 m^2 = k h^2 mn \implies m^2 + n^2 = kmn$, where $GCD(m, n) = 1$.
So without loss of generality we can consider the primitive $m^2 + n^2 = kmn$.
As $m \ne n$ they cannot both be equal to 1, so let us suppose that $m \gt 1$. In addition, $m$ must contain a prime factor that $n$ does not contain; call this prime factor, $p$.
Clearly $m^2$ and $kmn$ are divisible by $p$, but $n^2$ does not contain this factor. Hence no integer solution exists and we prove that $x^2 + y^2$ can never be multiple of $xy$.
If $x$, $y$, and $z$ are positive integers, does $x^2 + y^2 + z^2 = kxyz$ contain solutions?