Given a function of two positive integers $n_x^2+n_y^2$.
$n_x^2+n_y^2=50$ has three combinations of $n_x$ and $n_y$ that result in $n_x^2+n_y^2=50$:
$$n_x=7,n_y=1$$ $$n_x=5,n_y=5$$ $$n_x=1,n_y=7$$
I need to find the net highest integer such that there are three or more combinations of $n_x$ and $n_y$ that result in this number, but I have no clue how to go about it, other than by trial and error. Can anyone point me in the right direction? A hint would be preferable.
Allowing zeros and distinguishing signs and order, you could use the Sum of Two Squares Function $r_2(n)$. It is related to Jacobi's $\vartheta$ function: $\vartheta_3^2(q)$ is known as the generating function of $r_2(n)$.
So other than trail and error, you could Taylor expand $\vartheta_3^2(q)$ the orders $50+$ and check if the coefficient is higher than $3\cdot 4=12$, since pairs of squares are split into four, due to signs.