## Puzzle of sum to composite

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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
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
• ... 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

Nice Sunday to all of you,

Maximilian

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