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 $24$?

Solution 1:

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.