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• ... These aren t sets, they re multisets. ... This does not define a Matrix Addition table. Is 0 0 0 0 such a table? Is 1 4 9 16 such a table? ... Given that
Message 1 of 2 , Dec 16, 2006
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--- Kermit Rose <kermit@...> wrote:
> Matrix Factor Element Sets
>
> A Matrix Factor Element Set is defined recursively as follows.
>
> Any set of distinct non negative integers which includes zero is a
> Matrix Factor Element Set.
>
> If any positive integer within a Matrix Factor Element Set is replaced
> by two copies of
> that integer less 1, then an equivalent Matrix Element Set is created.

These aren't sets, they're multisets.

> Conversely, if two copies of an integer within a Matrix Factor Element
> Set is replaced by
> one copy of the next larger integer, then an equivalent Matrix Element
> Set is Created.
>
> Example: {0,1,3,4,6} is a Matrix Factor Element Set.
>
> Replace the 6 by 5,4,3,2,2 to get the equivalent Matrix Factor Element Set
>
> {0,1,2,2,3,3,4,4,5}
>
> The challenge is: Given the fundamental Matrix Factor Element Set,
> find an equivalent Matrix Factor Element Set that may be made into
>
> For our example,
> {0,1,2,2,3,3,4,4,5} is such an equivalent Matrix Factor Element Set.
>
> It's Matrix Addition table is
>
> 0 2 3
> 1 3 4
> 2 4 5
>
> A Matrix Addition table must have at least two rows and two columns.
> The number of rows need not be the same as the number of columns.

This does not define a Matrix Addition table.

Is
0 0
0 0
such a table?

Is
1 4
9 16
such a table?

> What would you estimate, for general preset Matrix Factor Element Sets,
> the complexity of this problem to be?

Given that you've not defined your terms, it's impossible.

> How does this relate to prime numbers?
>
> Theorem:
> Let p be a prime odd positive integer.
> Let W be the Matrix Factor Element Set which contains exactly the
> exponents in the
> base 2 representation of p,
>
> Then there does not exist a Matrix Factor Element Set, equivalent to W,

It remains to be seen.

Phil

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