Exploring the Collatz Conjecture: A Distributed Computing Approach

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Exploring the Collatz Conjecture A Distributed
Computing Approach

• Student Name Desmond McPhillips
• Student ID 13806604
• Project Supervisor Dr. Piyush Ojha
•  
• Course BSc Hons Computing Science
• Date Dec 2007.

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Overview

• Introduction
• What is the Collatz Conjecture?
• Literature Review
• Aims and Objectives
• Requirements analysis and specification
• Initial Design
• Implementation Strategy
• Project Management

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Introduction

• Mathematical Conjecture
• An Mathematical proposition which is speculated
  to be true but has not been proved.
• The Collatz Conjecture is one such proposition.
• First proposed by Lothar Collatz in 1937
• The conjecture has eluded and intrigued
  mathematicians
• Prize fund established to reward the first
  person(s) to solve the problem.

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Introduction

• Easy to state
• The conjecture itself is easy to state the
  following iterations of any positive integer will
  always end in the Xn 4, Xn1 2, Xn2 1,
  Xn3 4 cycle
• Xi 1 ½ Xi if Xi is even
• Xi 1 3 Xi 1 if Xi is odd
• Example initial X 6
• 3,10,5,16,8,4,2,1 8 iterations
• Important Numbers 3, 16 and 8
• (These are the statistics gathered, useful for
  mathematicians)

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Literature Review

• Work on the conjecture has been an on-going
  process with the majority of the breakthroughs
  coming quite recently.
• Oliveira e Silva May 2007, verified the range
  up to 258 1.
• Eric Roosendaal -- June 2007, verified the range
  up to 259 2.
• Both used computational, distributed approach.
• 1. http//www.ieeta.pt/tos/3x1.html
  2.http//www.ericr.nl/wondrous/index.html

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Aims and Objectives

• The proposed project is quite similar to the
  approaches taken above, an application to compute
  mass ranges of values and verify the conjecture.
• Why re-invent the wheel?
• Key Differences
• Operating System Independence
• Automatic Distribution
• Graphical Interface

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Aims and Objectives

• Operating System Independence
• The established approaches have been dependant on
  the operating system in which they were first
  developed.
• Proposed project
• Windows
• Unix
• Mac OS
• With a greater range of operating systems
  supported, the application can gain wider user
  contribution.

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Aims and Objectives

• Automatic Distribution
• No importance to Silva.
• Roosendaals was restrictive.
• The proposed project is to make the block
  acquisition an automatic process. Once a client
  has completed a range of number and verified it,
  the client is then supplied with another range to
  verify.

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Aims and Objectives

• Graphical Interface
• The proposed project will include a graphical
  interface to make the process more meaningful to
  the client, especially if the application is used
  when the system is idle and can be used as a
  screensaver.

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Requirements analysis and specification

• To make this possible, certain aspects need to be
  addressed, such as
• Server application
• Shall store information about clients
  (identifier, hardware/system environments).
• Shall store this information in a back end
  database (MySQL).
• Shall assign ranges for computation/verification
  by clients.
• Shall aggregate statistical data from
  computations returned by clients

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Requirements analysis and specification

• Client application
• Shall verify the conjecture on the ranges
  assigned to it and gather the statistical data.
• Shall be OS Independent .
• Shall run when the clients machine is idle
  (possibly as a screensaver).
• Shall report to the server on completion and
  request a new range to work on.

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Initial Design

• Below is a prototype of the client side
  application interface

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Implementation Strategy

• Development Language
• Ideal language given the time constraints and
  requirements would be Java JNI can be
  implemented to offset any computational overhead.
• Prototype
• Complete, see prototype GUI.
• Incremental development
• A series of beta releases on a 6-8 week basis
  before the final release in April 2008.

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Implementation Strategy

• Testing
• Due to the nature of the application, access to a
  wide range of computers is needed to fulfil a
  test condition Wombat Financial Software will be
  able to satisfy this requirement with access to
  their mass computer network and servers.

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Project Management

• Since I have decided upon Incremental
  development, I have outlined my release goals
• Beta release one 17th Dec 2007
• Beta release two 4th Feb 2008
• Beta release three 24th March 2008
• Final release and project report April 2008

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Summary

• Computational exploration on the Collatz
  Conjecture.
• Distributed approach to the problem.
• Improve on the current approaches.
• Gather statistical information can be used to
  aid mathematical proof.
• All aspects completed April 2008
• Any Questions.. ?