If $(\sqrt{2}+\sqrt{3})^{1000}$ is written in decimal notation, say, $a_ka_{k-1}\ldots a_0.b_1b_2 \ldots$, what are $a_0$ and $b_1$?
Define integer sequences $\{x_n\}$ and $\{y_n\}$ by $(\sqrt{2}+\sqrt{3})^{2n}=(5+2\sqrt{6})^n=x_n+y_n\sqrt{6}, \quad \mbox{for} \; \, n \geq 0.$ Thus, $\{x_n\}$ and $\{y_n\}$ satisfy the recurrence formula $x_0=1,\; y_0=0, \;\; x_{n+1}=5x_n+12y_n, \; y_{n+1}=2x_n+5y_n, \quad \mbox{for} \;\, n \geq 0.$
From the definition of the sequences, $(\sqrt{2}-\sqrt{3})^{2n}=x_n-y_n\sqrt{6},$ and hence $(\sqrt{2}+\sqrt{3})^{2n}+(\sqrt{2}-\sqrt{3})^{2n}=2x_{n}.$
Since $|\sqrt{2}-\sqrt{3}|<1$, we see that $a_0$ is $2x_{500}-1$ computed modulo $10$, and $b_1=9$. The sequence $\{(x_n,y_n) \bmod{10}\}$ is $\{(1,0),(5,2),(9,0),(5,8),\ldots\},$ where the first four ordered pairs repeat. Hence $a_0=2x_0-1=1.$