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
Pythagorean prime representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers "p" such that there exists a right triangle, with integer legs, whose hypotenuse has length . They are also exactly the prime numbers "p" such that there exists a right triangle with integer sides whose hypotenuse has length "p". For, if the triangle with legs "x" and "y" has hypotenuse length (with "x" > "y"), then the triangle with legs "x" − "y" and 2"xy" has hypotenuse length "p". Another way to understand this representation as a sum of two squares involves Gaussian integers, the complex