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 he announced the following two results for odd primes formula_4: He also wrote: In other words, if "p, q" are of the form 20"k" + 3 or 20"k" + 7, then "pq" = "x" + 5"y". Euler later extended this to the conjecture that Both Fermat's assertion and Euler's conjecture were established by Lagrange. A generalization of Fermat's theorem, the sum of two squares theorem, characterizes the integers (not necessarily prime) that can be expressed as the sum of two squares. They are exactly the integers in which each prime that is congruent to 3 mod 4 appears with an