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
Fermat's theorem on sums of two squares he announced the following two results for odd primes formula_4: He also wrote: In other words, if "p, q" are of the form 20"k" + 3 or 20"k" + 7, then "pq" = "x" + 5"y". Euler later extended this to the conjecture that Both Fermat's assertion and Euler's conjecture were established by Lagrange. A generalization of Fermat's theorem, the sum of two squares theorem, characterizes the integers (not necessarily prime) that can be expressed as the sum of two squares. They are exactly the integers in which each prime that is congruent to 3 mod 4 appears with an