Fibonacci Golden Nuggets

Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.
For this problem we shall be interested in values of $x$ for which $A_F(x)$ is a positive integer.
Surprisingly$\begin{align*}
A_F(\tfrac{1}{2})
&= (\tfrac{1}{2})\times 1 + (\tfrac{1}{2})^2\times 1 + (\tfrac{1}{2})^3\times 2 + (\tfrac{1}{2})^4\times 3 + (\tfrac{1}{2})^5\times 5 + \cdots \\
&= \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{2}{8} + \tfrac{3}{16} + \tfrac{5}{32} + \cdots \\
&= 2
\end{align*}$

The corresponding values of $x$ for the first five natural numbers are shown below.

$x$$A_F(x)$
$\sqrt{2}-1$$1$
$\tfrac{1}{2}$$2$
$\frac{\sqrt{13}-2}{3}$$3$
$\frac{\sqrt{89}-5}{8}$$4$
$\frac{\sqrt{34}-3}{5}$$5$

We shall call $A_F(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the $10$th golden nugget is $74049690$.
Find the $15$th golden nugget.

To solve the problem, we need to derive the $x$ that makes the series $A_F(x)$ a positive integer. As given, the infinite polynomial series is represented in terms of the Fibonacci sequence.

The equation for the Fibonacci sequence in the series is given by: $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.

It can be rewritten as : $A_F(x) = x + x^2 + x^3 (x+x^2) + x^4 (x^2+x^3) + x^5 (x^3+ x^4) + \dots $, which simplifies to: $A_F(x) = x + x^2 (1+ x) + x^4(1+ x) + …$

This series can be written in a simpler form using the formula for geometric series: Sum $= a / (1 – r)$, where $a$ is the first term and $r$ is the common ratio.

By setting $a = x$ and $r = x^3$, one can write the series as:

$A_F(x) = x / (1 – x^3)$

Equating the series to a positive integer, let’s say ‘n’, gives:

$x = n (1 – x^3)$, or $x^3n + x – n = 0$

This equation can be solved in terms of ‘n’ for the possible roots ‘x’, out of which the rational root must be found.

According to the question, the golden nugget value for a certain ‘n’ is the denominator of the simplified fraction form of ‘x’.

This cubic equation will result in one rational root and two irrational roots.

Given that the denominator for the first five natural numbers are 1, 2, 3, 8 and 5 respectively, it can be observed that the denominator forms a Fibonacci sequence, apart from the first term, when ‘n’ is increased sequentially.

So, the 15th denominator must equal the 14th Fibonacci number (since the first denominator corresponds to the second Fibonacci number).

The Fibonacci sequence starting from 3rd term (1, 2, 3, 5, 8…), the 14th term can be calculated. By a python program or otherwise, we find that the 14th term is 377.

This means that the 15th golden nugget is $377^2$ or $142129$, and that will be our answer.

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