Pythagorean theorem four elementary lemmata: Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. The proof is as follows: This proof, which appears in Euclid's "Elements" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. We have already discussed the Pythagorean proof, which was
Pythagorean prime Pythagorean prime A Pythagorean prime is a prime number of the form 4"n" + 1. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares. Equivalently, by the Pythagorean theorem, they are the odd prime numbers "p" for which is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers "p" for which "p" itself is the hypotenuse of a Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; is the hypotenuse of a