" tommy quaternion "

[tommy1729]

ok.

I need to add some important stuff.

there have basically been 3 strategies , and until now I mentioned only 2 :

The techniques are for multiplication of 2 distinct units in the multiplication table.

The techniques are basically equations satisfied most of the time.

All of them require commutative. And we want latin squares.
And non-associative.

Properties like power-associative , solvable , not nilpotent etc are the goal but not really part of the strategy ; we hope to get them as a bonus.

However we do use for the units :

(x y)^2 = x^2 y^2 = +/- 1.

The strategies are based on working without +/- signs first and then adding them later.




1) the first strategy is to use the quandle equations : 

a * (b * c) = (a * b) * (a * c)

a * b = b * a

a * a = a

and at the end (when the table is complete , but before adding the signs +/- )  replace a * a with +/-1 instead of a. 

I posted a pic of this before here in dimension 5.

But it turns out dimension needs to be a multiple of 4. (due to +/- and solvability )

2)

" Anti-associative "

a * (b * c) = - (a * b) * c

a * b = b * a

An example was given with dim 8 ; I posted a pic of that before.

3)

The mersenne structure.

We want the property A ( for the units at least )

x * (y * 1/y) = x = (x * y) * 1/y

so ignoring signs this implies and not counting the real part as dimension here : 

in dimension 3 : 

a b = c

a c = b

b c = a

Now it becomes clear that this "triangle structure" must always exist between any 3 units to satisfy property A.

The number of dimensions or equiv the number of unit variables is then restricted.

Lets see 

if we have x elements and add one element Y,

then we must multiply every one of the x elements with x , add the x elements and the one Y element.

So we end up mirroring the x elements with Y and adding the one Y.

In other words :

adding a single element and still requiring the " triangle structure " or equivalently property A , we must have 

2 x + 1 elements.

so the function 2x + 1 is a way to increase the dimension.

Also if dimension x and y exist , then so does xy , this follows simply from (x,y) meaning a sort of subgroup extentions.

example 

a b = c

a c = b

b c = a


leads to 


x_1 = (a,a) 
x_2 = (b,a)
x_3 = (c,a)
x_4 = (a,b)

etc

so we get x_9 elements and the products are similarly defined 

examples : 
(a,b) * (a,c) = (a,a)
(b,c) * (a,b) = (c,a)

So we end up with 2 function to construct higher dimensions :

2x + 1

and products 

xy

.

IF we start with 1 or 3 and only use iterations of 2x + 1 we get

1,3,7,15,...

those are the mersenne numbers : 2^n - 1.

and when they are mersenne primes they have no substructure.

But we also had the products method.

so for instance dimension 21 is possible :

(2^2 - 1) * (2^3 - 1) = 3 * 7 = 21.

It is a fun question to study what numbers can be reached and similar number theory and constructions.

( notice the slight analogue with collatz maybe : constructions with 2x and (x-1)/3 being able to make all integers > 1 is the collatz conjecture )

Ok so without the reals we can have 21 dimensions.

adding the reals we get 22 dimensions.

However 22 / 4 is not integer so it is not solvable.

That is the basic idea.

The smallest dimension that might work with strategy 3 ( being solvable aka multiple of 4 )  is then dimension 8.

Im investigating it.

3*3*3 + 1 = 28 
that is also a possible dimension ( 28 = 4 * 7 and 3*3*3 is a mersenne construction )
( the +1 is adding the real dimension )

Yes strategy 3 is my new favorite.

Im not sure if not nilpotent can be defended though.

Also I have not explicitly shown non-associative but that is pretty easy.
Non-associative is preserved by 2x + 1 and by the products xy , so by induction it is easy.

***

remarks

I want to add that I believe this is one of the reasons why there are infinitely many mersenne primes rather than infinitely many fermat primes.

more precisely , apart from this deep algebra ,

the equation

f(n+1) = a f(n) + b

with a and b relatively prime 

has no solutions of type 2^n + 1 or similar.

but it does have solutions like 2^n - 1 or (3^n - 1)/2 ...

***

regards

tommy1729

So our dimensions without the real dimensions are 

1,3,7,9,15,19,21,27,31,39,43,45,49,55,57,63,79,81,87,91,93,99,...

It is an interesting integer sequence.

Density theorems would be nice.

but that is a bit off topic.

regards

tommy1729