## Problem:

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.

210

## Code:The solution may include methods that will be found here: Library.java .

public interface EulerSolution{public String run();}
/*  * Solution to Project Euler problem 40 * By Nayuki Minase *  * http://nayuki.eigenstate.org/page/project-euler-solutions * https://github.com/nayuki/Project-Euler-solutions */public final class p040 implements EulerSolution {		public static void main(String[] args) {		System.out.println(new p040().run());	}			public String run() {		StringBuilder sb = new StringBuilder();		for (int i = 1; i < 1000000; i++)			sb.append(i);				int prod = 1;		for (int i = 0; i <= 6; i++)			prod *= sb.charAt(Library.pow(10, i) - 1) - '0';		return Integer.toString(prod);	}	}