Two sequences, So weird !

Algebra Level 5

Two real sequences {an}n=0 \displaystyle {\{a_{n}\}}_{n=0}^{\infty},{bn}n=0 \displaystyle {\{b_{n}\}}_{n=0}^{\infty} are defined as follows a1=α \displaystyle a_{1}=\alpha and b1=β \displaystyle b_{1}=\beta ,an+1=αanβbn \displaystyle a_{n+1}=\alpha a_{n}-\beta b_{n} bn+1=βan+αbnb_{n+1}=\beta a_{n}+\alpha b_{n} For all n2n \ge 2 . How many non-zero pairs (α,β)(\alpha,\beta) of real numbers are there such that a1997=b1 \displaystyle a_{1997}=b_{1} and b1997=a1 \displaystyle b_{1997}=a_{1}?

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