Fibonacci golden nuggets
January 12, 2007
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 = Fk−1 + Fk−2, 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) |
|---|---|
| √2−1 | 1 |
| 1/2 | 2 |
| (√13−2)/3 | 3 |
| (√89−5)/8 | 4 |
| (√34−3)/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.
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