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Topic: sum of squres
Replies: 3   Last Post: Jan 27, 2008 10:46 PM

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Vinod Tyagi

Posts: 6
Registered: 12/8/04
sum of squres
Posted: Jan 15, 2008 2:19 AM
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In his recently published paper, 'The Degen- Graves- Caley Eight Square Identity',in wikipedia.org, July 16,2005, Titus Piezas III presents an overview of certain beautiful algebraic identities involving sum of squares primarily of the form

(x12+...+xn2 ) (y12+...+yn2 ) = (z12 +...+zn2).

Starting with the famous Brahmagupta-Fibonacci identity on sum of two squares, Lebesgue's three square identity and Euler's four square identity, the author presented Degen's Eight Square Identity and even beyond. For details of this paper, please log on to wikepedia.

In this small note, we present a conjecture on sum of squares which I have not been able to prove in general (And that I have not been able to locate on any internet site), according to which

Conjecture: If two nonzero integers N and M can be expressed as a sum of n squares, then their product NM can be expressed as sum of n squares in at least n-different ways.

We give below four examples for n up to 5 for a set of randomly chosen integers to substantiate our argument.

Example 1. For n= 2; let N = 5 = 12 + 2 2 and M = 25 = 32 +42, then their product 125 can be expressed as
125 = 102 +52
= 112 +22.

Example 2. For n = 3; let N= 9 = 12 + 2 2 +22 and M = 14 = 12 + 2 2 + 32 ,then their product 126 can be expressed as
126 = 112 +22 +12
= 102 +52 +12
= 92 + 62 +32 .

Example 3. For n = 4; let N = 30 = 12 + 2 2 + 32 +42 and 174 = 52 + 62 + 72 +82,then their product 5220 can be expressed as sum of four squares in atlest four different ways as follows:

5220 = 712 + 132 + 32 +12
= 692 + 152 + 152 +32
= 682 + 242 + 42 + 22
= 672 + 592 + 252 + 52

Example 4. For n = 5; Let N = 55 =12 +2 2 +32 +42 + 52 and M =255 = 52 + 62 + 72 +82 +92 then their product 14025 can be expressed as sum of five different squares in five different ways as follows:

14025 = 1172 +102 + 102 + 102 + 62
= 1162 +202 + 122 + 42 + 32
= 1152 + 242 + 122 + 82 + 42
= 1142 + 302 + 102 + 52 +22
= 1132 + 302 + 162 + 82 + 62
etc.
I hope this result will be of some interest to people working on number of squares.



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