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Puzzle of sum to composite

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  • Kermit Rose
    Hello Friends. Prove that every positive composite integer can be expressed as a sum of 4 positive integers such that the product of the 4 positive integers is
    Message 1 of 5 , Apr 15, 2012
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      Hello Friends.

      Prove that every positive composite integer can be expressed as a sum of
      4 positive integers
      such that the product of the 4 positive integers is a square integer.

      Kermit
    • Maximilian Hasler
      ... N=ab=(p+q)(r+s)=pr+ps+qr+qs pr * ps * qr * qs = ( pqrs )^2 otherwise said, /any/ decomposition of the two factors of N into a sum leads to a decomposition
      Message 2 of 5 , Apr 15, 2012
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        On Sunday, April 15, 2012, Kermit Rose <kermit@...> wrote:
        > Prove that every positive composite integer can be expressed as a sum of
        > 4 positive integers
        > such that the product of the 4 positive integers is a square integer.


        N=ab=(p+q)(r+s)=pr+ps+qr+qs
        pr * ps * qr * qs = ( pqrs )^2

        otherwise said, /any/ decomposition of the two factors of N into a sum
        leads to a decomposition satisfying your constraints.

        Nice Sunday to all of you,

        Maximilian


        [Non-text portions of this message have been removed]
      • Robert Gerbicz
        You can also generalize the problem for non-composites, the only exceptions are N=1,2,3,5, for the other cases these are good decompositions:
        Message 3 of 5 , Apr 15, 2012
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          You can also generalize the problem for non-composites, the only exceptions
          are N=1,2,3,5, for the other cases these are good decompositions:
          2n=1+1+(n-1)+(n-1)
          2n+1=1+4+(n-2)+(n-2)


          [Non-text portions of this message have been removed]
        • WarrenS
          ... --every positive integer is a sum of 4 squares (Fermat, Euler, Lagrange); the product of 4 squares is a square. QED. Note the word composite was not
          Message 4 of 5 , Apr 15, 2012
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            --- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
            >
            > Hello Friends.
            >
            > Prove that every positive composite integer can be expressed as a sum of
            > 4 positive integers
            > such that the product of the 4 positive integers is a square integer.
            >
            > Kermit


            --every positive integer is a sum of 4 squares (Fermat, Euler, Lagrange);
            the product of 4 squares is a square.
            QED.

            Note the word "composite" was not required.
          • Robert Gerbicz
            ... Missed the condition that here the 4 terms should be positive, and in the Lagrange s theorem you can use 0 as a term. [Non-text portions of this message
            Message 5 of 5 , Apr 15, 2012
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              2012. �prilis 16. 2:00 WarrenS �rta, <warren.wds@...>:

              > **
              >
              >
              >
              >
              > --- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
              > >
              > > Hello Friends.
              > >
              > > Prove that every positive composite integer can be expressed as a sum of
              > > 4 positive integers
              > > such that the product of the 4 positive integers is a square integer.
              > >
              > > Kermit
              >
              > --every positive integer is a sum of 4 squares (Fermat, Euler, Lagrange);
              > the product of 4 squares is a square.
              > QED.
              >
              > Note the word "composite" was not required.
              >
              >
              >
              Missed the condition that here the 4 terms should be positive, and in the
              Lagrange's theorem you can use 0 as a term.


              [Non-text portions of this message have been removed]
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