Problem 140 of Project Euler is very much a continuation of the Problem 137, as we can see from the problem description
Consider the infinite polynomial series AG(x) = xG1 + x2G2 + x3G3 + …, where Gk is the kth term of the second order recurrence relation Gk = Gk-1 + Gk-2, G1 = 1 and G2 = 4; that is, 1, 4, 5, 9, 14, 23, … .
For this problem we shall be concerned with values of x for which AG(x) is a positive integer.
The corresponding values of x for the first five natural numbers are shown below.
x AG(x) (√5-1)/4 1 2/5 2 (√22-2)/6 3 (√137-5)/14 4 1/2 5 We shall call AG(x) a golden nugget if x is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.
Find the sum of the first thirty golden nuggets.
In Problem 137 I mentioned in the end that the problem could be solved using a Diophantine equation. This is exactly the way I will go for this problem. Continue reading →