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Binet's formula provides a proof that a positive integer {{mvar|x}} is a Fibonacci number [[if and only if]] at least one of <math>5x^2+4</math> or <math>5x^2-4</math> is a [[Square number|perfect square]].<ref>{{Citation | title = Fibonacci is a Square | last1 = Gessel | first1 = Ira | journal = [[The Fibonacci Quarterly]] | volume = 10 | issue = 4 | pages = 417–19 |date=October 1972 | url = https://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf | access-date = 2012-04-11 }}</ref> This is because Binet's formula, which can be written as <math>F_n = (\varphi^n - (-1)^n \varphi^{-n}) / \sqrt{5}</math>, can be multiplied by <math>\sqrt{5} \varphi^n</math> and solved as a [[quadratic equation]] in <math>\varphi^n</math> via the [[quadratic formula]]:
<math display=block>\varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^{\!2} + 4(-1)^n}}{2}.</math>
Comparing this to <math>\varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2</math>, it follows that
: <math display=block>5{F_n}^{\!2} + 4(-1)^n = (F_n + 2F_{n-1})^2\,.</math>
In particular, the left-hand side is a perfect square.
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