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Define a Fibonacci sequence by

$$\varphi_0=0,\,\varphi_1=1;\ \varphi_{n+2}=\varphi_{n+1}+\varphi_n\ \forall \,n\in\mathbb Z^+\cup\{0\}.$$

Show that

$$5\varphi_n^2+4(-1)^n$$

is a perfect square for all non-negative integers $n$.

$$\varphi_0=0,\,\varphi_1=1;\ \varphi_{n+2}=\varphi_{n+1}+\varphi_n\ \forall \,n\in\mathbb Z^+\cup\{0\}.$$

Show that

$$5\varphi_n^2+4(-1)^n$$

is a perfect square for all non-negative integers $n$.

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