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
Fermat's theorem on sums of two squares modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625. The statement that every prime "p" of the form "4n+1" is the sum of two squares is sometimes called "Girard's theorem". For his part, Fermat wrote an elaborate