I think that Problem 137 of Project Euler is a really fantastic problem since it has so many facets of how it can be solved. I will go through a one of them, and then link to a few other. The problem reads

Consider the infinite polynomial series A_{F}(x) =xF_{1}+x^{2}F_{2}+x^{3}F_{3}+ …, where F_{k}is thekth 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 ofxfor 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 ofxfor the first five natural numbers are shown below.

xA_{F}(x)√2-111/22(√13-2)/33(√89-5)/84(√34-3)/55

We shall call A_{F}(x) a golden nugget ifxis rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.

Find the 15th golden nugget.