Fibonacci sequence
Identification

Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of 5 x 2 + 4 {\displaystyle 5x^{2}+4} or 5 x 2 − 4 {\displaystyle 5x^{2}-4} is a perfect square. This is because Binet's formula, which can be written as F n = (φ n − (− 1) n φ − n) / 5 {\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}}, can be multiplied by 5 φ n {\displaystyle {\sqrt {5}}\varphi ^{n}} and solved as a quadratic equation in φ n {\displaystyle \varphi ^{n}} via the quadratic formula:

φ n = F n 5 ± 5 F n 2 + 4 (− 1) n 2. {\displaystyle \varphi ^{n}={\frac {F_{n}{\sqrt {5}}\pm {\sqrt {5{F_{n}}^{2}+4(-1)^{n}}}}{2}}.}

Comparing this to φ n = F n φ + F n − 1 = (F n 5 + F n + 2 F n − 1) / 2 {\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}=(F_{n}{\sqrt {5}}+F_{n}+2F_{n-1})/2}, it follows that

In particular, the left-hand side is a perfect square.
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