$If both $n$ and $\sqrt{n^2+204n}$ are positive integers, find the maximum value of $𝑛$.$

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If both $n$ and $\sqrt{n^2+204n}$ are positive integers, find the maximum value of $n$.

Number Theory Math Competitions Algebra

Paul F

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Let
\[?^2=?^2+204?\]
\[?^2=?^2+204?+10404=(?+102)^2.\]
Then
\[(???)(?+?)=2^2?3^2?17^2 \]
Since $???$ and $?+?$ have the same parity, they must be both even. Let
\[\frac{???}{2}\frac{?+?}{2}=3^2?17^2\]
Since $?>?$ there are only two possibilities: $???=2$ and $???=18$. The former gives $?=2600$ and the latter $?=280$. Thus we are interested in the former, which gives $?=2500$.

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$If both $n$ and $\sqrt{n^2+204n}$ are positive integers, find the maximum value of $𝑛$.$

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