h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Regarding the sequence: -

n' = 3n + 1 - for all even n.
n' = n / 2 - for all odd n.

Hopefully you've all realised this sequence converges towards 1 as soon as it hits a power of 2 (i.e. 2 ^ m - where m is integral).

You may also have noticed that the number it hits to begin convergence is infact an even power of 2 (i.e. 2 ^ 2m - where m is integral). This is due to the fact that even powers of 2 yield a remainder of 1 when divided by 3.

Concentrating on odd values of n only, we have: -

n' = 3n + 1 - where n is odd.

The final odd number in the sequence will satisfy the equation: -

n' = 2 ^ 2m - where m is even.

Therefore: -

3n + 1 = 2 ^ m
3n = 2 ^ m - 1
n = ( 2 ^ m - 1 ) / 3 - where n is odd and m is even.

Expressed another way: -

3( 2n + 1 ) + 1 = 2 ^ 2m
6n + 4 = 2 ^ 2m
6n = 2 ^ 2m - 4
n = ( 2 ^ 2m - 4 ) / 6 - where n and m are integral.

Having taken it this far, I'm completely lost and I've forgotten my original idea. Why am I even writing this? What is my purpose? I sure as hell don't know, but I've written this much and it's a shame to waste it, so I'll leave it for someone else to pick up where I left off if anyone has the faintest idea where I was going with this...

Cheers.

Another convergence happens when the number is of the form 10*(2^n), since this will eventually reduce to 5, which leads to the series 16, 8, 4, 2, 1.

Actually, strictly speaking, this is a never-ending series. It simply reaches an endless loop of 1, 4, 2, 1 etc.

I suspect the proof may lie in modular arithemetic. As noted by others, every odd number in the sequence is always followed by an even number. However even numbers can be followed by either an even or an odd number, but the last digit is restricted i.e.

A number ending in a

0 will lead to a number ending in 0 or 5
1 " " " " " " " 4
2 " " " " " " " 1 or 6
3 " " " " " " " 0
4 " " " " " " " 2 or 7
5 " " " " " " " 6
6 " " " " " " " 3 or 8
7 " " " " " " " 2
8 " " " " " " " 4 or 9
9 " " " " " " " 8

Unfortuntely this is a area of mathematics I haven't studied, so I'm at a loss how to take it forward.