Ruud H.G. van Tol on Tue, 04 Jan 2022 04:31:58 +0100

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Re: Collatz nature

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.

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

-- Ruud