Pythagorean theorem Pythagorean theorem In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides "a", "b" and "c", often called the "Pythagorean equation": where "c" represents the length of the hypotenuse and "a" and "b" the lengths of the triangle's other two sides. Although it is
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