Pythagoras triangle the square of the hypotenuse is equal [to the sum of] the squares of the two other sides"—that is, formula_1. According to a popular legend, after he discovered this theorem, Pythagoras sacrificed an ox, or possibly even a whole "hecatomb", to the gods. Cicero rejected this story as spurious because of the much more widely held belief that Pythagoras forbade blood sacrifices. Porphyry attempted to explain the story by asserting that the ox was actually made of dough. The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, but it is possible that he
Proofs of Fermat's theorem on sums of two squares has two involutions: an obvious one formula_213 whose fixed points formula_214 correspond to representations of formula_82 as a sum of two squares, and a more complicated one, which has exactly one fixed point formula_217. Two involutions over the same finite set must have sets of fixed points with the same parity, and since the second involution has an odd number of fixed points, so does the first. Zero is even, so the first involution has a nonzero number of fixed points, any one of which gives a representation of formula_82 as a sum of two squares. This proof, due to