By Wikimedia user Cmglee, a visual proof that 3³ + 4³ + 5³ = 6³:

This value, 216, is sometimes called Plato’s number because it seems to correspond to this enigmatic passage in the Republic:

Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births.

Unfortunately, Plato’s meaning is far from certain. Other values that have been proposed (with various rationales) include 1,728, 3,600, 5,040, 8,128, 17,500, 760,000, and 12,960,000. Cicero politely described the question as “obscure”; it remains open.