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The 3n+1 Conjecture - Proof Needed!

Many things in mathematics are very easy to state but very hard to prove. The 3n+1 Conjecture is a little known example of this. It has been proposed by a number of different people, including Professor Bryan Thwaites; he has offered £1,000 to anyone who can prove or disprove it. So far, no-one has claimed the prize. Would you like to have a go?

Easy statements and lack of proof seems to apply most of all in the theory of numbers, which deals with the counting numbers 1, 2, 3 etc. Some problems are notorious and their solution provides the mathematician with instant fame. One such example of this is Fermat's Last Theorem.

The 3n+1 Conjecture in a Nutshell

Take any positive whole number. If it is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process over and over again. Eventually you will reach 1.

For example, start with 3. This goes through the numbers 3, 10, 5, 16, 8, 4, 2 and 1. Since it has reached 1, we stop.

It doesn't matter what number you start with, you will always end up at 1. Nobody knows if this is true for all numbers or not, but it is certainly true for those numbers that have been tested. Try it for a few numbers. There is a strange fascination in doing this and seeing how high the number gets before it starts to head back down towards 1. The number may grow and shrink many times, without any apparent pattern, until eventually it reaches the final resting point.

Here's an example of a number that takes quite a while: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

If anybody out there comes up with a proof that it is true, or an example of a number which never reaches 1, they will achieve a small amount of fame in the world of mathematics, plus a cheque for £1,000 in their pocket. Pencils at the ready ...

This conjecture appears to have been proposed independently by a number of people, including Professor Bryan Thwaites and Lothar Collatz. As a result, it goes under a few different names: the Thwaites Conjecture, the 3n+1 Conjecture, the Syracuse problem, the Collatz Conjecture and the Kakutani Conjecture.

The following are links to other sites which discuss this conjecture:

Pure vs Applied Mathematics

The 3n+1 conjecture is an example of pure mathematics. It has no known application. There may well be an application out there waiting to be discovered but that is not the reason mathematicians pursue such problems. In the words of one Researcher:

Some very interesting and useful mathematics can often be invented (or is it discovered?) while working on 'useless' problems. Of course, that's not the motivation for those who dedicate years of their lives to proving such conjectures, as I have. We know that we are looking directly at faces of the mind of God that man has never seen before, and that's enough to justify our lives, and hopefully to keep us away from the bottle for a while...

Who said 'useless'?

 Proof given. (Last Posting: Jun 6, 2011) I think the problem is unprovable. (Last Posting: Aug 9, 2010) Any odd number will do! (Last Posting: May 1, 2009) Thwaite's conjecture (Last Posting: Sep 13, 2005) A few references... (Last Posting: Dec 17, 2003) Thwaites proof?? (Last Posting: Oct 1, 2010) Contacting Prof. Thwaites 2009 (Last Posting: Jan 8, 2010) I have made a program (Last Posting: Dec 15, 2006) I'd like to quote this (Last Posting: May 9, 2005) Some Comments about the Collatz conjecture (Last Posting: Jul 12, 2003)

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Entry Data
Entry ID: A565788 (Edited)

Written and Researched by:
U151503 - Gnomon - Future Guide Editor - towels at the ready
GTBacchus

Edited by:
Tom I.

Date: 07   June   2001

 Referenced Guide Entries Fermat's Last Theorem Pencils

 Referenced Sites nrich - Specialists in Rich Mathematics CECM

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Most of the content on h2g2 is created by h2g2's Researchers, who are members of the public. The views expressed are theirs and unless specifically stated are not those of the BBC. The BBC is not responsible for the content of any external sites referenced. In the event that you consider anything on this page to be in breach of the site's House Rules, please click here. For any other comments, please start a Conversation above.