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Old 2007-07-21, 18:28   #1
nibble4bits
 
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Nov 2005

2·7·13 Posts
Default Related to Collatz conjecture

Define a function that can only return integer results, that must always follow the pattern of the next value being 1/3rd, or the next value being 1 plus twice the current value.

If x/3 === 0 (mod 3), then x'=x/3
Else x'=2x+1

There are some interesting trees/cycles one can make by changing those coefficients! I'm thinking that there are artist applications of this idea, as well as the purely mathematical.

The formula could be generalized as:
If (x mod a) == 0, then x' = x/a
Else x'=bx+c

Each combination of a,b, and c leads to different graphs.


Another interesting question has to do with the running fractions based on x/a and bx+c. Is there a method for specific combinations of values, to prove they always stop or at least stop in less steps than the initial x?


*Bonus* Extend the formula to the following table for x mod a (start at 0, end at a-1):
x/a
bx+c
dx+e
fx+g
...

Last fiddled with by nibble4bits on 2007-07-21 at 18:29
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