# The Sum of Perfect Squares

In Symmetry and the Monster, I ran across this interesting fact:

Let $\displaystyle f(n) = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \sum_{k=1}^{n} k^2$

Let $x$ be an integer

Then $f(n) = x^2$ for only two tuples $(n,x)$: $(1,1)$ and $(24,70)$

How would you prove this? Intuitively, is there something special about $24$?

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The identity $f(24) = 70^2$ can be explained with the Leech lattice but it isn't clear to me how this helps proving that there is no other such identity. –  t.b. May 3 '11 at 15:53
It can also be found using elliptic curves, also there is an elementary proof but it is two pages of hard work. –  quanta May 3 '11 at 15:58