Step by Step solution about How to resolve the algorithm Euler's sum of powers conjecture step by step in the PHP programming language

This code implements a solution to Project Euler's problem 30, which is to find the smallest number that can be written as the sum of fifth powers of at least four different numbers.

The code defines a function, eulers_sum_of_powers(), which calculates the smallest such number.

The function iterates over all possible combinations of four distinct powers of 5, checks if the sum of these powers is a fifth power, and if so, returns the four powers and the fifth power.

The code then calls eulers_sum_of_powers() and prints the result. The code also includes a function, pow(), which calculates the power of a number.

The complexity of the code is O(n^4), where n is the maximum value of n for which the code calculates the fifth powers.

<?php

function eulers_sum_of_powers () {
	$max_n = 250;
	$pow_5 = array();
	$pow_5_to_n = array();
	for ($p = 1; $p <= $max_n; $p ++) {
		$pow5 = pow($p, 5);
		$pow_5 [$p] = $pow5;
		$pow_5_to_n[$pow5] = $p;
	}
	foreach ($pow_5 as $n_0 => $p_0) {
		foreach ($pow_5 as $n_1 => $p_1) {
			if ($n_0 < $n_1) continue;
			foreach ($pow_5 as $n_2 => $p_2) {
				if ($n_1 < $n_2) continue;
				foreach ($pow_5 as $n_3 => $p_3) {
					if ($n_2 < $n_3) continue;
					$pow_5_sum = $p_0 + $p_1 + $p_2 + $p_3;
					if (isset($pow_5_to_n[$pow_5_sum])) {
						return array($n_0, $n_1, $n_2, $n_3, $pow_5_to_n[$pow_5_sum]);
					}
				}
			}
		}
	}
}

list($n_0, $n_1, $n_2, $n_3, $y) = eulers_sum_of_powers();

echo "$n_0^5 + $n_1^5 + $n_2^5 + $n_3^5 = $y^5";

?>