Pythagorean theorem four elementary lemmata: Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's "Elements" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. We have already discussed the Pythagorean proof, which was
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