## Problem:

Consider quadratic Diophantine equations of the form:

x2 – Dy2 = 1

For example, when D=13, the minimal solution in x is 6492 – 13[×]1802 = 1.

It can be assumed that there are no solutions in positive integers when D is square.

By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:

32 – 2[×]22 = 1
22 – 3[×]12 = 1
92 – 5[×]42 = 1
52 – 6[×]22 = 1
82 – 7[×]32 = 1

Hence, by considering minimal solutions in x for D [≤] 7, the largest x is obtained when D=5.

Find the value of D [≤] 1000 in minimal solutions of x for which the largest value of x is obtained.

1366

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

public interface EulerSolution{public String run();}
/*  * Solution to Project Euler problem 16 * By Nayuki Minase *  * http://nayuki.eigenstate.org/page/project-euler-solutions * https://github.com/nayuki/Project-Euler-solutions */import java.math.BigInteger;public final class p016 implements EulerSolution {		public static void main(String[] args) {		System.out.println(new p016().run());	}			public String run() {		String temp = BigInteger.ONE.shiftLeft(1000).toString();		int sum = 0;		for (int i = 0; i < temp.length(); i++)			sum += temp.charAt(i) - '0';		return Integer.toString(sum);	}	}