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
Legendre's three-square theorem Legendre's three-square theorem In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers if and only if is not of the form formula_2 for integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as formula_2) are Pierre de Fermat gave a criterion for numbers of the form 3"a" + 1 to be a sum of three squares essentially equivalent to Legendre's theorem, but did not provide a proof. N. Beguelin noticed in 1774 that every positive