data/problems/137.yml
---
:id: 137
:name: Fibonacci golden nuggets
:url: https://projecteuler.net/problem=137
:content: |+
Consider the infinite polynomial series A<sub>F</sub>(_x_) = _x_F<sub>1</sub> + _x_<sup>2</sup>F<sub>2</sub> + _x_<sup>3</sup>F<sub>3</sub> + ..., where F<sub><i>k</i></sub> is the _k_th term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, F<sub><i>k</i></sub> = F<sub><i>k</i>−1</sub> + F<sub><i>k</i>−2</sub>, F<sub>1</sub> = 1 and F<sub>2</sub> = 1.
For this problem we shall be interested in values of _x_ for which A<sub>F</sub>(_x_) is a positive integer.
| Surprisingly A<sub>F</sub>(1/2) | = | (1/2).1 + (1/2)<sup>2</sup>.1 + (1/2)<sup>3</sup>.2 + (1/2)<sup>4</sup>.3 + (1/2)<sup>5</sup>.5 + ... |
| | = | 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ... |
| | = | 2 |
The corresponding values of _x_ for the first five natural numbers are shown below.
| **_x_** | **A<sub>F</sub>(_x_)** |
| √2−1 | 1 |
| 1/2 | 2 |
| (√13−2)/3 | 3 |
| (√89−5)/8 | 4 |
| (√34−3)/5 | 5 |
We shall call A<sub>F</sub>(_x_) a golden nugget if _x_ is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.
Find the 15th golden nugget.