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Re: [PrimeNumbers] Matrix Factor Element Sets and Matrix Addition Table.

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  • 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 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
      > 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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