Solution to Project Euler Problem 46: Goldbach's other conjecture - It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square. 9 = 7 + 2×12 15 = 7 + 2×22 21 = 3 + 2×32 25 = 7 + 2×32 27 = 19 + 2×22 33 = 31 + 2×12 It turns out that the conjecture was false. What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

• Java Code

package sikademy;
/**
 *
 * @author Archangel Macsika
 * Copyright (c) Sikademy. All rights reserved
 */

public class SikademyEulerSolution {
	
	public String run() {
		for (int i = 9; ; i += 2) {
			if (!satisfiesConjecture(i))
				return Integer.toString(i);
		}
	}
	
	
	private static boolean satisfiesConjecture(int n) {
		if (n % 2 == 0 || isPrime(n))
			return true;
		
		for (int i = 1; i * i * 2 <= n; i++) {
			if (Library.isPrime(n - i * i * 2))
				return true;
		}
		return false;
	}

	public static boolean isPrime(int x) {
		if (x < 0)
			throw new IllegalArgumentException("Negative number");
		if (x == 0 || x == 1)
			return false;
		else if (x == 2)
			return true;
		else {
			if (x % 2 == 0)
				return false;
			for (int i = 3, end = sqrt(x); i <= end; i += 2) {
				if (x % i == 0)
					return false;
			}
			return true;
		}
	}

        public static int sqrt(int x) {
		if (x < 0)
			throw new IllegalArgumentException("Square root of negative number");
		int y = 0;
		for (int i = 1 << 15; i != 0; i >>>= 1) {
			y |= i;
			if (y > 46340 || y * y > x)
				y ^= i;
		}
		return y;
	}

	public static void main(String[] args) {
                SikademyEulerSolution solution = new SikademyEulerSolution();
		System.out.println(solution.run());
	}	
}

• Python Code

# 
# @author Archangel Macsika
# Copyright (c) Sikademy. All rights reserved.
# 

import itertools

def compute():
	ans = next(itertools.filterfalse(test_goldbach, itertools.count(9, 2)))
	return str(ans)


def test_goldbach(n):
	if n % 2 == 0 or is_prime(n):
		return True
	for i in itertools.count(1):
		k = n - 2 * i * i
		if k <= 0:
			return False
		elif is_prime(k):
			return True

def sqrt(x):
	assert x >= 0
	i = 1
	while i * i <= x:
		i *= 2
	y = 0
	while i > 0:
		if (y + i)**2 <= x:
			y += i
		i //= 2
	return y

# Tests whether the given integer is a prime number.
def is_prime(x):
	if x <= 1:
		return False
	elif x <= 3:
		return True
	elif x % 2 == 0:
		return False
	else:
		for i in range(3, sqrt(x) + 1, 2):
			if x % i == 0:
				return False
		return True

if __name__ == "__main__":
	print(compute())