It's interesting that no one has so far mentioned Pythagorean triples and Fermat's Last Theorem :)
EDIT: With sufficient imagination any integer can be interpreted as the dimension of an appropriate vector space. However, in this particular case hardly any imagination is necessary: it is in dimensions 1 and 2 only that there exists 3 hypercubes with integer sides such that the sum of volumes of the first two is equal to the volume of the third one.