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There have been many results about the number of squares needed to represent a positive integer. Lagrange's four-square theorem tells us that $4$ squares suffice for any integer and there have been results showing when you can do better than that.

What I'm interested in is why people consider these results interesting.

I can think of one interesting question that relates to these results. If we are allowed to construct lengths by drawing lines with integer lengths and making right-angled triangles out of them to construct new lengths the above theorem tells us that it is always possible to construct a line of length $\sqrt n$ (for any natural number $n$) and it is always possible to do that in at most $4$ "steps".

But beyond that this strikes me as a pretty random result. I'd therefore be interested in additional motivations for this result - historic or modern.

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Like other important theorems in number theory, this one relates the additive structure of the integers with its multiplicative structure. – lhf Aug 25 '13 at 12:29
Thanks. How exactly does it do that though? – Timotej Aug 25 '13 at 12:46
"square" is a multiplicative concept, and "sum" is an additive concept --- that's how! – Gerry Myerson Aug 25 '13 at 13:28
up vote 4 down vote accepted

I believe that originally (e.g. when Fermat discovered that only the primes congruent to $1$ modulo $4$ could be written as sums of squares) this was seen mainly as a "curiosity".

From a more modern point of view, the problem of representing an integer as a sum of squares is part of the general theory of quadratic forms. An important step in the classification of quadratic forms over, say, a number field $K$ is the characterization of the elements of $K$ represented by a given non-degenerate quadratic form $Q(x_1,..,x_n)$, i.e. the elements $k\in K$ for which there exist (non-trivial) solutions of the equation $$ Q(x_1,..,x_n)=k. $$ When $n=2$ this generalizes the problem of finding the norm subgroup in a quadratic extension of $\Bbb Q$ (which is linked to Class Fields Theory) and in general has an obvious significance in terms of arithmetic geometry of quadrics and Brauer groups.

Moreover, the number of representations (i.e. how many solutions are there) has a deeper significance as Fourier coefficients of certain automorphic forms, see Serre's A Course in Arithmetic for an introduction to these ideas.

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