--- 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

> a Matrix Addition Table.

>

> 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,

> which can be made into a Matrix Addition Table.

It remains to be seen.

Phil

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