# Two sequences, So weird !

Algebra Level 5

Two real sequences $$\displaystyle {\{a_{n}\}}_{n=0}^{\infty}$$,$$\displaystyle {\{b_{n}\}}_{n=0}^{\infty}$$ are defined as follows $$\displaystyle a_{1}=\alpha$$ and $$\displaystyle b_{1}=\beta$$ ,$\displaystyle a_{n+1}=\alpha a_{n}-\beta b_{n}$ $b_{n+1}=\beta a_{n}+\alpha b_{n}$ For all $$n \ge 2$$ . How many non-zero pairs $$(\alpha,\beta)$$ of real numbers are there such that $$\displaystyle a_{1997}=b_{1}$$ and $$\displaystyle b_{1997}=a_{1}$$?

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