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Shlomi Fish > Math-GrahamFunction > Math::GrahamFunction



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Module Version: 0.02002   Source  


Math::GrahamFunction - Calculate the Graham's Function of a Natural Number.


version 0.02002


    use Math::GrahamFunction;

    my $calc = Math::GrahamFunction->new({ 'n' => 500 });

    my $results = $calc->solve();

    print "The Graham Function of 500 is ",


The Graham Function of a natural number n, which we will denote by G(n), is the minimal number for which there's an increasing series of integers that starts at n and ends at G(n) whose product is a perfect square.

This module calculates the Graham Function of a natural number, along with the entire associated series.


On 11 December 2002, Mark Jason Dominus gave a Perl Quiz-of-the-week challenge to write a Perl program to calculate the Graham Function. I came up with a solution for it, whose complexity was polynomial (as opposed to brute force solutions, which are exponential complexity.). This module is derived from my original code, after it was heavily refactored.

More information about the algorithm and the original code can be found here:


version 0.02002


my $calc = Math::GrahamFunction->new({'n' => $n});

Initializes a new object for solving the Graham's Function of the number $n. Call solve() next.

my $results = $calc->solve();

Calculates the Graham's Function series for the number (could be time consuming), and returns a hash ref of results. The only field of interest there is 'factors', which points to an array reference of the series. The series is increasing so $results-{factors}->[0]> is $n and $results-{factors}->[-1]} is the Graham's Function.


Get the Square factors of the number $number.


Shlomi Fish, <shlomif at>


The module may yield different sequences with its "factor in between" optimization than without it. The last number (= the Graham function) is the same, but the numbers in between are different. A future release will provide a flag to disable that optimization.


Please report any bugs or feature requests to bug-math-grahamfunction at, or through the web interface at I will be notified, and then you'll automatically be notified of progress on your bug as I make changes.


You can find documentation for this module with the perldoc command.

    perldoc Math::GrahamFunction

You can also look for information at:


The latest source for this module is available from its subversion repository:


Mark Jason Dominus ( ) for the original Graham Function Quiz-of-the-Week.

imacat ( and David Golden for helping me debug a CPAN smoking failure with installing this module on imacat's computer.


Copyright 2007 Shlomi Fish, all rights reserved.

This program is released under the following license: MIT X11.

Note: the module meta-data says this module is released under the BSD license. However, MIT X11 is the more accurate license, and "bsd" is the closest option for the CPAN meta-data.


Shlomi Fish <>


This software is Copyright (c) 2018 by Shlomi Fish.

This is free software, licensed under:

  The MIT (X11) License


Please report any bugs or feature requests on the bugtracker website

When submitting a bug or request, please include a test-file or a patch to an existing test-file that illustrates the bug or desired feature.



You can find documentation for this module with the perldoc command.

  perldoc Math::GrahamFunction


The following websites have more information about this module, and may be of help to you. As always, in addition to those websites please use your favorite search engine to discover more resources.

Bugs / Feature Requests

Please report any bugs or feature requests by email to bug-math-grahamfunction at, or through the web interface at You will be automatically notified of any progress on the request by the system.

Source Code

The code is open to the world, and available for you to hack on. Please feel free to browse it and play with it, or whatever. If you want to contribute patches, please send me a diff or prod me to pull from your repository :)

  git clone
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