\section*{Problem 21}

For which positive integers \(n\) is \(3^{2 n+1}-2^{2 n+1}-6^{n}\) composite?

Answer \(\quad\) all \(n \neq 1\)


\section*{Solution}

\(3^{2 n+1}-2^{2 n+1}-6^{n}=\left(3^{n}-2^{n}\right)\left(3^{n+1}+2^{n+1}\right)\), so it is certainly composite for \(n>1\). For \(n=1\), it is \(27-8-6=13\), which is prime.

