Exploring the Collatz Conjecture: A Distributed Computing Approach

Student Name: Desmond McPhillipsStudent ID: 13806604

Project Supervisor: Dr. Piyush Ojha

Course: BSc Hons Computing Science Date: Dec 2007.

Overview

• Introduction: What is the Collatz Conjecture?

• Literature Review• Aims and Objectives• Requirements analysis and specification• Initial Design• Implementation Strategy• Project Management

IntroductionMathematical 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.

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, Xn+1 = 2, Xn+2 = 1, Xn+3 = 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)

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 2^58 1.

• Eric Roosendaal -- June 2007, verified the range up to 2^59 2.

• Both used computational, distributed approach.

1. http://www.ieeta.pt/~tos/3x+1.html 2.http://www.ericr.nl/wondrous/index.html

Aims and ObjectivesThe 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

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.

Aims and Objectives

Automatic Distribution:

• No importance to Silva.

• Roosendaal’s 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.

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.

Requirements analysis and specification

To make this possible, certain aspects need to be addressed, such as:

• Server applicationShall 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

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.

Initial Design

Below is a prototype of the client side application interface:

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.

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.

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

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.. ?