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
Fermat's theorem on sums of two squares he announced the following two results for odd primes formula_4: He also wrote: In other words, if "p, q" are of the form 20"k" + 3 or 20"k" + 7, then "pq" = "x" + 5"y". Euler later extended this to the conjecture that Both Fermat's assertion and Euler's conjecture were established by Lagrange. A generalization of Fermat's theorem, the sum of two squares theorem, characterizes the integers (not necessarily prime) that can be expressed as the sum of two squares. They are exactly the integers in which each prime that is congruent to 3 mod 4 appears with an