2. Find the numbers $a$ and $n$ knowing that $a$ is a prime number, $n \in \mathbb{N}^{*}$, and $a^{2 n}-5=4\left(5+5^{2}+\cdots+5^{2015}\right)$.

Prof.Vlad Emil, Zimnicea
2. $a^{2 n}=5+4 \cdot 5+4 \cdot 5^{2}+\cdots+4 \cdot 5^{2015}=5(1+4)+4 \cdot 5^{2}+\cdots+4 \cdot 5^{2015}=$ $=5^{2}(1+4)+4 \cdot 5^{3}+\cdots+4 \cdot 5^{2015}=\cdots=5^{2015}+4 \cdot 5^{2015}=5^{2016}$. $4 p$ Therefore $a^{2 n}=5^{2016}, a$ prime $\Rightarrow a=5, n=1008$ $3 p$