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Matrix Factor Element Sets and Matrix Addition Table.

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  • Kermit Rose
    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
    Message 1 of 2 , Dec 16, 2006
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      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.

      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.

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


      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.

      Kermit < kermit@... >
    • Phil Carmody
      ... 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 2 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
        > 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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