I remember vaguely that the equation $1^2+\dots+24^2=70^2$ gives a construction of the Leech lattice (one has probably to consider the even neighbour  of the lattice $\mathbb Z \frac{w}{70}+\Lambda'$ where $w=(1,2,\dots,24)$ and $\Lambda'=\lbrace z\in\mathbb Z^{24}|\langle z,w\rangle\in70\mathbb Z\rbrace$) which is a rather unique and exceptional structure.