Solve in integers the equation $x^2 + y^2 - 1 = 4xy$
Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$
for some natural number $k < n$.
Solve in $\textit{rational}$ numbers the equation $x^2 - dy^2 = 1$ where $d$ is an integer.