Duncan's blog

December 18, 2014

Project Euler: problem 46 – Goldbach’s other conjecture

Filed under: PHP,Project Euler — duncan @ 11:57 pm
Tags: , , ,

46I’m doing these Project Euler mathematical puzzles as a simple practical exercise for teaching myself PHP, and I’d appreciate any feedback on my code

Problem 46:

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?

This was another of those problems that I’d initially passed over, having decided it looked a bit tricky.  Then looking at it again realised it probably wasn’t too difficult.  And I turned out to be right!

A composite number is basically any positive integer that isn’t a prime number.

My logic is simply:

  • Loop through odd numbers, from 3 upwards.
  • For each number, if it’s prime, add it to an array of primes for later reference.
  • If it’s not prime, work out if it matches the conjecture:
    • Subtract the next largest prime, then examine the remainder
    • Divide the remainder by 2.  if it’s a square number then we’re meeting the conjecture.  Move onto the next odd number.
    • Otherwise keep subtracting the primes, checking the remainders.
    • If we’ve looped through all the primes then we must have reached a number that doesn’t meet the conjecture.

This runs in about 40ms:

$primes = [2];

$i= 1;

while (true) {
	$i+= 2;

	if (isPrime($i)) {
		$primes[] = $i;
	} else {
		for($j = count($primes)-1; $j > 0; $j--) {
			$remainder  = $i - $primes[$j];
			$remainder = $remainder / 2;
			$root = sqrt($remainder);
			if ((int) $root == $root) {
				continue 2;
		echo $i;

function isPrime($x)
	$root = sqrt($x);
	for ($i = 3; $i <= $root; $i += 2) {
		if ($x % $i == 0) {
			return false;
	return true;

I’m using a modified version of my original isPrime function, just because I know I don’t need to check for even numbers.

The loop backwards through the primes was maybe a bit of overkill; using a simple foreach loop through the primes in ascending order took more like 100ms.

What else… I check if a square root is the same as when it’s cast to an integer (not sure this is the best approach).  Then use continue 2; to get out of our inner-most loop and move onto the next value in our parent loop.

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