Monday, August 16, 2010

About Lothar

Lothar Collatz was born 6 July 1910 in Amsberg, Westphalia.  During his post-doctoral studies at at the age of 27, Collatz developed what is now known as the Collatz Conjecture.

Simply stated, the Collatz Conjecture is defined in terms of the predicted properties of a rudimentary function.

We define Collatz's function, f, as:

f(n) = \begin{cases} \frac{n}{2} & \text{if $n \equiv 0$ (mod 2)} \\ 3 n + 1 & \text{if $n \equiv 1$ (mod 2)} \end{cases}

for any positive starting integer n, the conjecture states that repeated iterations of this function will always lead to the cycle { 4, 2, 1 }.  In other words, repeatedly applying this function to some starting number will in the long run cause the number
to become smaller and that there are no other cycles in the sequence besides { 4, 2, 1 }.  To this day, no-one has been able to fully prove this conjecture holds true for all natural numbers but clearly it has for every number as yet tested.  Do you think you can find the exception that will prove Lothar wrong?

The inverse function, f-1, is defined as the sequence moving backward from an ending integer through to some initial starting integer that would eventually arrive at the number entered through a series of forward iterations. It is defined as follows:

f^{-1}(n) = \begin{cases} 2 n & \text{if $n \equiv 0, 1, 2, 3, 5$ (mod 6)} \\ \frac{n - 1}{3} & \text{if $n \equiv 4$ (mod 6) \& ``Previous'' pressed} \\ 2 n & \text{if $n \equiv 4$ (mod 6) \& ``Multiply'' pressed} \end{cases}

This application was written by Jeffrey C. Jacobs and is Copyright ©2010 TimeHorse, LLC; source code is available upon request.


To use Lothar, you enter a number into the edit box using the keypad provided and then hit the next to precede through successive values in the Collatz numeric sequence.  To follow the sequence in reverse, activate the reverse switch and click the previous button through successive iterations.  Occasionally, when viewing the previous numeric values in the Collatz sequence, a multiply button will appear.  Where as normally this number could have been arrived at by taking an odd number, multiplying by 3 and adding 1, it's also possible that number was arrived at by dividing some number by 2.  By default, previous always selects the first option; select multiply to select the second.

Clicking on the i will take you to the about screen where you can learn more about the Collatz algorithm and about the man who first described it, Lothar Collatz.  The about page is a built-in web browser with forward, back and reload buttons.  When a page is loading, the reload button becomes as stop button to stop the action.  There is also a home button to take you back to the original about screen.  Note the about page is stored within the application and does not need to use the Internet to retrieve data.  However, clicking links on the about page will load external data to the applications web browser.

If you have any support questions, please ask them here and I shall try to address them as quickly as possible.

No comments:

Post a Comment