In Number Theory, Logic, and Cryptography

Collatz Conjecture


Take any positive integer: if the number is even, divide it by two; if the number is odd, triple it and add one (for example, if this operation is performed on 26, the result is 13; if it is performed on 5, the result is 16).  Perform this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. 

For example, starting at 11, the sequence goes 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

The Collatz conjecture is: this process will eventually reach the number 1, regardless of which positive integer is chosen initially.

The problem is to prove the conjecture, or find a counter-example.


For further information, please see:

[1] http://mathworld.wolfram.com/CollatzProblem.html

[2] http://en.wikipedia.org/wiki/Collatz_conjecture

[3] http://www-personal.ksu.edu/~kconrow/


 You can check for contributions to this problem on the solutions page.



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Tim S Roberts

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