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Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$.

How to prove or disprove it? I am unable to get any idea on it. It would be of great help if any one could help.

E. Joseph
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  • I am not sure if my comment is right or wrong, but if this were possible we could have invented a $3$ dimensional number system, just like the complex number forms a $2$ dimensional number system. I would like someone who is more mathematically mature than me to clarify this comment. –  Dec 07 '13 at 21:35

1 Answers1

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It is always a good idea to try some simple examples to get a feel for what is going on:

$$3 = 1^2+1^2+1^2$$

and

$$5 = 2^2+1^2+0^2$$

but it is easy to verify that 15 is not a sum of 3 squares. In fact any number of the form $8n+7$ cannot be a sum of 3 squares, as can be seen from looking at simple modulus possibilities.

Old John
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  • I am not sure if my comment is right or wrong, but if this were possible we could have invented a $3$ dimensional number system, just like the complex number forms a $2$ dimensional number system. I would like someone who is more mathematically mature than me to clarify this comment. –  Dec 07 '13 at 21:36
  • @user17762 I might know just the man: Will Jagy knows much more than I do about such things. – Old John Dec 07 '13 at 21:39
  • Ok, then let me catch his attention here. @WillJagy I am not sure if my comment is right or wrong, but if this were possible we could have invented a 3 dimensional number system, just like the complex number forms a 2 dimensional number system. Of course, the reason for my question is due to the following: Product of two number, each of which can be written as a sum of two squares is again a sum of two squares. This is directly related to the fact that $\vert z_1 z_2 \vert = \vert z_1 \vert \vert z_2 \vert$, where $z_1,z_2 \in \mathbb{C}$. –  Dec 07 '13 at 21:41
  • @user17762 and something similar happens with a product of 2 numbers which are sums of 4 squares - directly related to quaternions (now near the limit of my understanding). – Old John Dec 07 '13 at 21:46
  • @user17762, see http://en.wikipedia.org/wiki/Division_algebra#Not_necessarily_associative_division_algebras and http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28normed_division_algebras%29 and http://en.wikipedia.org/wiki/Hurwitz_problem as well as pages 127-131 in T. Y. Lam (2005), Introduction to Quadratic Forms over Fields. – Will Jagy Dec 07 '13 at 22:27