Find all positive integers $n$ such that $12n-119$ and $75n-539$ are both perfect squares.

Solution

Let,

\begin{align} 12n-119 &&=&& x^{2} \label{y2012r1p4e1}\\ 75n-539 &&=&& y^{2} \label{y2012r1p4e2} \end{align}

From $\ref{y2012r1p4e1}$ and $\ref{y2012r1p4e2}$, we have

\begin{align} 4y^{2}-25x^{2} = (2y-5x)(2y+5x) = 3 \cdot 273=9 \cdot 91 = 39 \cdot 21 = 13 \cdot 63 \label{y2012r1p4e3} \end{align}

Solving the sets of simultaneous equations resulting from $\ref{y2012r1p4e3}$ and restricting ourselves to integer solutions, we get the following solutions $(x,y)=(27,69)$ and $(x,y)=(5,19)$.

Using $\ref{y2012r1p4e1}$ and $x=5$ we get $n=12$. For $x=27$, $n$ is not an integer.