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Numerically I find the positive integer solution of the equation $F_n=5x^2+7$, where $F_n$ denotes the $n^\text{th}$ Fibonacci number, as $(n,x)=(16,14)$ and I guess that the only positive solution of it is $(n,x)=(16,14)$.

How can I solve this equation?

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    It's known that $5F_n^2+4(-1)^n$ is a square. Putting in $F_n=5x^2+7$ we get the equations $y^2=125x^4+350x^2+241$ and $y^2=125x^4+350x^2+249$ (depending on the parity of $n$). There are well-understood techniques for solving such equations.2012-07-30

2 Answers 2