Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Proofs of Fermat's theorem on sums of two squares at least one such partition exists if formula_107 is congruent to 1 modulo 4. Proofs of Fermat's theorem on sums of two squares Fermat's theorem on sums of two squares asserts that an odd prime number "p" can be expressed as with integer "x" and "y" if and only if "p" is congruent to 1 (mod 4). The statement was announced by Girard in 1625, and again by Fermat in 1640, but neither supplied a proof. The "only if" clause is easy: a perfect square is congruent to 0 or 1 modulo 4, hence a sum of two squares is