## Fibonacci golden nuggets

### Problem 137

Published on Friday, 12th January 2007, 06:00 pm; Solved by 2613

Consider the infinite polynomial series AF(x) = xF1 + x2F2 + x3F3 + ..., where Fk is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, Fk = Fk1 + Fk2, F1 = 1 and F2 = 1.

For this problem we shall be interested in values of x for which AF(x) is a positive integer.

Surprisingly AF(1/2) | = | (1/2).1 + (1/2)2.1 + (1/2)3.2 + (1/2)4.3 + (1/2)5.5 + ...
| = | 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 | AF(x)
21 | 1
1/2 | 2
(132)/3 | 3
(895)/8 | 4
(343)/5 | 5

We shall call AF(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.