A theorem of Lagrange states that any natural number can be written as a sum of this many (*) squares of integers.
This is the only dimension for which there exist differential manifolds which are homeomorphic but not diffeomorphic to Euclidean space. This is the smallest number such that there are two nonisomoprhic groups of this order, since the Klein group has this order but is not cyclic. A theorem of Lagrange states that any natural number can be written as a sum of this many (*) squares of integers. This number is the order of the largest complete planar graph, and Appel and Haken proved that this number is the maximum chromatic number for a planar graph. The Abel-Ruffini theorem states that this is the highest order of a polynomial equation that can be solved by radicals. For 10 points, a famous theorem states that any map coloring needs only how many colors?
This is the only dimension for which there exist differential manifolds which are homeomorphic but not diffeomorphic to Euclidean space.
four