Elementary algebra sign). One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle: This equation states that formula_44, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by formula_45 and formula_46. An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as formula_47); such equations are called
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