Pythagorean theorem three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by Pappus of Alexandria in 4 AD The
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