Re: Collatz nature

["Ruud H.G. van Tol" <rvtol@isolution.nl>]

On 2022-01-04 03:43, Ruud H.G. van Tol wrote:
> The real value of the Collatz conjecture,
> is that it shows us that it is useful
> to extend the number system, on the logarithm side,
> with a cyclic dimension, or, more specifically,
> to the (addition, multiplication) tuple,
> very similar to how useful imaginary numbers turned out to be.
>
> So the structure exposed by the Collatz conjecture should not be proven,
> but rather be explored, and put to use.
> It simply needs to be taken as a distinctive nature
> of the product of multiplication and addition.
>
> Logarithms are one dimension of joining addition with multiplication,
> and it has its inverse, Exponentation.
> "(3x+1)" with "(x-1)/3" are very much like that,
> so they need names too, without references to the involved digits.
> Variants like Ligarithm/Expinent are probably too ugly and confusing.
> So come up with better ones!
>
> Adding a cyclic dimension,
> similar to what gave us imaginary numbers,
> and certainly not limited to the "123" in "(3x+1)/2",
> is the "new math" side of it.

"(3x+1)/2"
a = 3
b = 1, b < a
c = 2, c < a

Define '[ x ]' as: divide `x` by `c` until congruent to `b` mod `c`.

Define '{ x }' as: multiply `x by `c` until congruent to `b` mod `a`
(so `x` can not be an `a`-fold, unless `b` is zero).

[ 20 ] = 5
{ 5 } = 10, 40, 160, ...

x -> (3x+1)/2
p3:_9|<7|11|17|13|>5|_1|
p2:__|28|22|34|26|10|_2|
_____|__|__|__|52|20|_4|
_____|__|__|__|__|40|_8|
_____|__|__|__|__|__|16|


-- Ruud