## Fibonacci golden nuggets

### Problem 137

Published on Friday, 12th January 2007, 06:00 pm; Solved by 3222; Difficulty rating: 50%Consider the infinite polynomial series A_{F}(*x*) = *x*F_{1} + *x*^{2}F_{2} + *x*^{3}F_{3} + ..., where F_{k} is the *k*th term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; 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 A_{F}(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 | A_{F}(x) |

√2−1 | 1 |

1/2 | 2 |

(√13−2)/3 | 3 |

(√89−5)/8 | 4 |

(√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 10th golden nugget is 74049690.

Find the 15th golden nugget.