The Collatz Problem

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A function T maps integers to integers.

Definition of T:
T(n) = n/2 if n is even
T(n) = (3n + 1)/2 if n is odd.

Collatz problem: Given any starting integer N if one keeps apply the function T to the result one eventually ends up with 1. T(N). T(T(N)) etc eventually gets to 1. Regardless of N.

A ten year old keep could understand this, but showing that T eventually produces 1 regardless of the starting numbers has eluded mathematicians for about 70 years. Pault Erdos once said that mathematicians are not yet ready to solve the Collatz problem.

ruveyn

Ow, my brain.

So, if I understand that correctly, no matter what number you have as N, if you plug it into either of those functions enough times, you eventually end up with 1? Interesting.

It's really one function, defined piecewise. Take n. If it is even, T(n) = n/2. If it is odd, T(n) = (3n+1)/2.

A function that is simply T(n) = n/2 is a different beast. Any n not a power of 2 would not go to 1.

But yes, with the function the OP gave, every n tried so far has gone to 1. But nobody has yet proven that all n goes to 1.

That should be T(n) = 3n + 1 if n is odd.

Interesting, though.

It does not matter. Once the odd n is transformed to 3n + 1, the next step is to divide it by 2.

ruveyn

Last edited by ruveyn on 05 Oct 2011, 6:23 pm, edited 1 time in total.

Slightly modified for a parity sequence, as I understand it - because when n is odd, 3n + 1 is always even.
The Hailstone parity sequence for every number then eventually enters the cycle 0 - 1 - 0.
The basic function is just 'HOPTO' - Half Or Triple Plus One, with the Hailstone sequence eventually reaching 4 - 2 - 1.

Even if they solved it, what could they possibly use the solution for? Does it have any application?

Sometimes the journey is more useful or more interesting than the destination.
It's fascinating just because it is. Makes one ask "Huh? How come?" - like a crab canon: http://strangepaths.com/canon-1-a-2/2009/01/18/en/

Many 'pure math' problems have turned out to have real world applications.

It doesn't matter to a human being. It matters to a machine.