257. Find several integer solutions of the equation $x^{2}$ $-3 y^{2}=1$, knowing the solution: $x=2, y=1$.
257. We have $x+y \sqrt{3}=(2+\sqrt{3})^{k} ; \quad$ taking $\quad k=2$, we find $x+y \sqrt{3}=7+4 \sqrt{3}$, from which $x=7, y=4$. Next, $x=4 \cdot 7-2=26, y=4 \cdot 4-1=15 ; \quad x=4 \cdot 26$ $-7=97, y=4 \cdot 15-4=56$ and so on. (see No. 256).