Have I correctly derived an inverse to the Binet formula?

Question

I was interested by another user's question on finding such an inverse and in particular noted Will Orrick's comment in the best answer that one can square both sides to obtain a quartic.I thought I'd solve that quartic. For clarity, here is the work in full; the customary notation $F_n$ has been replaced with $F$ for simplicity. $\phi$ is the golden ratio $\frac{1 + \sqrt{5}}{2}$, $n$ is the position in the sequence, and $F$ is the $n$th Fibonacci number.

$$ \begin{align} n &= \log_\phi\frac{F\sqrt{5} + \sqrt{5F^2 + 4(-1)^n}}{2}\\ \phi^n &= \frac{F\sqrt{5} + \sqrt{5F^2 + 4(-1)^n}}{2}\\ 2\phi^n &= F\sqrt{5} + \sqrt{5F^2 + 4(-1)^n}\\ 2\phi^n - F\sqrt{5} &= \sqrt{5F^2 + 4(-1)^n}\\ 4\phi^{2n} - 4\phi^nF\sqrt{5} + 5F^2 &= 5F^2 + 4(-1)^n\\ 4\phi^{2n} - 4\phi^nF\sqrt{5} &= 4(-1)^n\\ \phi^{2n} - \phi^nF\sqrt{5} &= (-1)^n\\ \phi^{4n} - 2\phi^{3n}F\sqrt{5} + 5\phi^{2n}F^2 &= 1\\ \phi^{4n} - 2\phi^{3n}F\sqrt{5} + 5\phi^{2n}F^2 - 1 &= 0 \end{align} $$

At this point, we can apply the quartic formula (edit: for clarity I am showing all the steps of applying the formula):

$$a = 1, b = 2F\sqrt{5}, c = 5F^2, d = 0, e = -1\\ \begin{align} u &= \frac{3(2F\sqrt{5})^2 - 8(1)(5F^2)}{12(1^2)} = \frac{60F^2 - 40F^2}{12} = {5F^2}{3}\\ \Delta_0 &= (5F^2)^2 - 3(2F\sqrt{5})(0) + 12(1)(-1) = 25F^4 - 12\\ \Delta_1 &= 2(5F^2)^3 - 9(2F\sqrt{5})(5F^2)(0) + 27(2F\sqrt{5})^2(-1) - 72(5F^2)(-1) = 250F^6 - 180F^2\\ Q &= \sqrt[3]{\frac{250F^6 - 180F^2 + \sqrt{(250F^6 - 180F^2)^2 - 4(25F^4 - 12)^3}}{2}} = \sqrt[3]{125F^6 - 90F^2 + 6\sqrt{48 - 75F^4}}\\ v &= \frac{Q^2 + \Delta_0}{3Q}\\ w &= 0 \end{align} $$

$$n_{1,2} = \log_\phi \frac{1}{2}\left(F\sqrt{5} - \sqrt{\frac{5F^2}{3} + \frac{\left(\sqrt[3]{125F^6 - 90F^2 + 6\sqrt{48 - 75F^4}}\right)^2 + 25F^4 - 12}{3\sqrt[3]{125F^6 - 90F^2 + 6\sqrt{48 - 75F^4}}}}\\ \pm \sqrt{\frac{10F^2}{3} - \frac{\left(\sqrt[3]{125F^6 - 90F^2 + 6\sqrt{48 - 75F^4}}\right)^2 + 25F^4 - 12}{3\sqrt[3]{125F^6 - 90F^2 + 6\sqrt{48 - 75F^4}}}}\right)$$

The third and fourth solutions are given by the first expression, but with a plus sign after $F\sqrt{5}$.

Are my calculations correct (they were done entirely by hand), is this a correct inverse to the Binet formula, and if so, how do the multiple solutions correspond to the actual solutions of n for F?

Answer 1

I follow all the steps up to line seven of the first part. If we write $x=\phi^n$ then that seventh line reads $$x^2-xF\sqrt{5}=(-1)^n.$$ Then on line eight you "square both sides", the goal being to eliminate the $(-1)^n$ on the right side. But suppose instead of actually distributing the square of the left side, you just keep it algebraically. Then the following line is equivalent to your ninth line (and so must have the same roots, and be the same quartic polynomial): $$ [1] \ \ [x^2 - xF\sqrt{5}]^2-1=0.$$ This is where your first paragraph ends, and you begin applying the solution of the quartic equation. But that is to use a more general tool than necessary for your purpose, since in fact [1] is a difference of squares, so that it factors into two quadratic polynomials. In other words the quartic you have is really a biquadratic, and its solutions are those of the two quadratic factors (two from each, for the total of four). Specifically [1] factors as $$[2]\ \ [x^2-xF\sqrt{5}-1]\cdot [x^2 - x F\sqrt{5}+1]=0.$$ Now depending on whether the original $n$ is odd or even, exactly one of these factors will have one of its roots lead to an integer for $n$, while the other root of that factor will be in a way conjugate to it (but won't lead to an integer $n$). The two quadratic roots of the other factor will not lead to integer $n$ in terms of $F$ either. In sum, only one of the four roots of your quartic will be the desired inverse for $F$, and the quadratically expressed roots of [2] must coincide with the four more convoluted expressions you have obtained using the solution for general quartics.