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#!/usr/bin/perl
#
# TASK #2 - Cardano Triplets
#
# Write a script to generate first 5 Cardano Triplets.
#
# A triplet of positive integers (a,b,c) is called a Cardano Triplet if
# it satisfies the below condition.
#
# cuberoot(a+b.sqrt(c)) + cuberoot(a-b.sqrt(c)) = 1
#
# Example
#
# (2,1,5) is the first Cardano Triplet.
#
# MY NOTES: Ok, two mildly tricky things:
# 1. real arithmetic means "=" is imprecise, we can't even use rationals as
# sqrt() and cuberoot() are involved, and
# 2. (2,1,5) is the "first" Cardano triplet - in what order?
#
# The answer to the latter question will set the basic structure
# of the program. Perhaps we should "order by minimum sum of triplet numbers"?
#
use strict;
use warnings;
use feature 'say';
use Getopt::Long;
use Data::Dumper;
my $debug=0;
my $epsilon = 1e-7;
#
# my $root = cuberoot($x);
# Return the cuberoot of $x; make sure it works even if $x<0
#
sub cuberoot ($)
{
my( $x ) = @_;
my $r = abs($x)**(1/3);
$r = -$r if $x<0;
return $r;
}
#
# my $isc = is_cardano($a,$b,$c);
# Return 1 iff $a,$b,$c is a Cardano triplet; 0 otherwise.
# Recall that a Cardano triplet has:
# cuberoot(a+b.sqrt(c)) + cuberoot(a-b.sqrt(c)) = 1
#
sub is_cardano ($$$)
{
my( $a, $b, $c ) = @_;
my $x = $b * sqrt($c);
my $q = cuberoot($a+$x) + cuberoot($a-$x);
#say "iscard($a,$b,$c): x=$x, q=$q";
return abs( $q-1 ) <= $epsilon ? 1 : 0;
}
#say is_cardano(2,1,5);
#exit 0;
die "Usage: first-N-cardano-triplets [--debug] [N default 5]\n" unless
GetOptions( "debug"=>\$debug ) && @ARGV<2;
my $n = shift // 5;
my $nfound = 0;
for( my $sum=3; $nfound < $n; $sum++ )
{
# find all Cardano triplets whose sum is $sum
say "find all Cardano triplets with sum $sum" if $debug;
for( my $a=1; $a<$sum-1; $a++ )
{
for( my $b=1; $a+$b<$sum; $b++ )
{
my $c = $sum-$a-$b;
die "bad structure, a=$a, b=$b, c=$c\n"
if $c<1;
#say "$a,$b,$c";
next unless is_cardano($a,$b,$c);
$nfound++;
say "Found $a,$b,$c";
}
}
}
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