Question 125
Suppose positive integers $m$, $n$, $k$ satisfy $mn = k^2 + k + 3$. Prove that at least one of the following Diophantine equations
$$
x^2 + 11y^2 = 4m \text{ and } x^2 + 11y^2 = 4n
$$
has a solution $(x, y)$ with $x$, $y$ being odd numbers.
$$
x^2 + 11y^2 = 4m \text{ and } x^2 + 11y^2 = 4n
$$
has a solution $(x, y)$ with $x$, $y$ being odd numbers.