\section*{Problem 1 - 201021}

Determine all natural numbers \(n\) for which the number \(1+4 \cdot 9^{2 n}\) is a prime number!
}

It holds that

\[
1+4 \cdot 9^{2 n} \equiv 1+4 \cdot(-1)^{2 n} \equiv 1+4 \equiv 0 \quad(\bmod 5)
\]

Therefore, \(1+4 \cdot 9^{2 n}\) is only prime for \(n=0\).