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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
up vote 8 down vote accepted

Here is an article that proves this fact (hopefully it is accessible). Also, there just usually aren't many numbers that are more than one kind of figurate number at once, and 24 happens to be the only solution when we look for square pyramidal numbers that are also square numbers.

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Here is JSTOR link to the article: jstor.org/stable/2323911 – Martin Sleziak Jan 20 at 8:22

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