<?php
/*
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 * 1^2
15 = 7  + 2 * 2^2
21 = 3  + 2 * 3^2
25 = 7  + 2 * 3^2
27 = 19 + 2 * 2^2
33 = 31 + 2 * 1^2
 
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?
*/

$primes = [2];
$composites = [];
$twicesquares = [];

function is_prime(int $n) :bool{for($i=$n**.5|1;$i&&$n%$i--;);return!$i&&$n>1;}

for ($i= 3; $i<6000; $i += 2) {
    if (is_prime($i))
    {
        $primes[] = $i;
    }
    else
    {
        $composites[] = $i;
    }
}

for($i=1; $i<100; $i++)
{
    $twicesquares[] = 2*($i*$i);
}

foreach($composites as $composite)
{
    $count = 0;
    foreach($primes as $prime)
    {
        if($prime > $composite)
        {
            break;
        }
        if(in_array($composite-$prime, $twicesquares))
        {
            $count++;
            break;
        }
    }

    if($count == 0)
    {
        echo "Found $composite\n";
        die;
    }
}