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
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