1953
Problem - 2955
Let $n$ be a positive integer, and $d$ is a positive divisor of $2n^2$. Show that $(n^2+d)$ cannot be a square number.
Let $2n^2 = kd$ where $k$ is a positive integer. If the conclusion is not true, then let $n^2 +d = x^2$ where $x$ is a positive integer. Then, we have $$k^2x^2 = k^2(n^2+d)=n^2(k^2+2k)$$ which cannot hold because $k^2 + 2k$ cannot be a square.