**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.

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.

**Solution:**

210

**Code:**

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

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);

}

}

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