How to resolve the algorithm Euler's sum of powers conjecture step by step in the Perl programming language

Published on 12 May 2024 09:40 PM
#Perl

How to resolve the algorithm Euler's sum of powers conjecture step by step in the Perl programming language

Table of Contents

Problem Statement

There is a conjecture in mathematics that held for over two hundred years before it was disproved by the finding of a counterexample in 1966 by Lander and Parkin. This conjecture is called Euler's sum of powers conjecture and can be stated as such: In 1966, Leon J. Lander and Thomas R. Parkin used a brute-force search on a CDC 6600 computer restricting numbers to those less than 250. The task consists in writing a program to search for an integer solution of

x

0

5

x

1

5

x

2

5

x

3

5

=

y

5

{\displaystyle x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}=y^{5}}

where all

x

i

{\displaystyle x_{i}}

and

y

{\displaystyle y}

are distinct integers between 0 and 250 (exclusive). Show an answer here. Related tasks are:

Let's start with the solution:

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

Source code in the perl programming language

use constant MAX => 250;
my @p5 = (0,map { $_**5 } 1 .. MAX-1);
my $s = 0;
my %p5 = map { $_ => $s++ } @p5;
for my $x0 (1..MAX-1) {
  for my $x1 (1..$x0-1) {
    for my $x2 (1..$x1-1) {
      for my $x3 (1..$x2-1) {
        my $sum = $p5[$x0] + $p5[$x1] + $p5[$x2] + $p5[$x3];
        die "$x3 $x2 $x1 $x0 $p5{$sum}\n" if exists $p5{$sum};
      }
    }
  }
}


use constant MAX => 250;
my @p5 = (0,map { $_**5 } 1 .. MAX-1);
my $rs = 5;
for my $x0 (1..MAX-1) {
  for my $x1 (1..$x0-1) {
    for my $x2 (1..$x1-1) {
      my $s2 = $p5[$x0] + $p5[$x1] + $p5[$x2];
      $rs-- while $rs > 0 && $p5[$rs] > $s2;
      for (my $x3 = 1;  $x3 < $x2;  $x3++) {
        my $e30 = ($x0 + $x1 + $x2 + $x3 - $rs) % 30;
        $x3 += (30-$e30) if $e30;
        last if $x3 >= $x2;
        my $sum = $s2 + $p5[$x3];
        $rs++ while $rs < MAX-1 && $p5[$rs] < $sum;
        die "$x3 $x2 $x1 $x0 $rs\n" if $p5[$rs] == $sum;
      }
    }
  }
}

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