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 appears in article 182 of the Disquisitiones Arithmeticae. An (integral binary) quadratic form is an expression of the form formula_105 with formula_106 integers. A number formula_107 is said to be "represented by the form" if there exist integers formula_108 such that formula_109. Fermat's theorem on sums of two squares is then equivalent to the statement that a prime formula_82 is represented by the form formula_111 (i.e., formula_112, formula_113) exactly when formula_82 is congruent to formula_91 modulo formula_116. The discriminant of the quadratic form is defined to be formula_117. The discriminant of formula_111 is then equal to formula_119. Two forms formula_120