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**EinStone** Let $\displaystyle x,y,d \in \mathbb{N}$. For fixed $\displaystyle d > 0$ we want to find solution pairs $\displaystyle (x,y)$ to the equation $\displaystyle x^2-dy^2=1$.

Assume we have already found the solution $\displaystyle (x_1,y_1)$ so that any other solution $\displaystyle (x,y)$ has $\displaystyle y_1 < y$.

Show that any pair $\displaystyle (x_k,y_k)$ satisfying $\displaystyle x_k + y_k\sqrt{d} = (x_1 + y_1\sqrt{d})^k$ for $\displaystyle k > 1$ is another solution.