The side lengths are said to form a Pythagorean triple. They are always different
integers, the smallest of them being at least 3.
of a certain pair of distinct positive integers , ; the
other cathetus is simply .
If there is given the value of as the length of the
hypotenuse and a compatible value as the length of one
cathetus, the pair of equations (1) and (2) does not determine
the numbers and uniquely (cf. the Proposition 4 in the
entry integer contraharmonic means). For example, if
and , then the equations give for
either or .
As for the hypotenuse and (1), the proof is found in  and also in the PlanetMath article contraharmonic means and Pythagorean hypotenuses. The contraharmonic and the harmonic mean of two integers are simultaneously integers (see this article (http://planetmath.org/IntegerHarmonicMeans)). The above claim concerning the catheti of the Pythagorean triangle is evident from the identity
If the catheti of a Pythagorean triangle are and , then the values of the parameters and determined by the equations (1) and (2) are
as one instantly sees by substituting them into the equations.
- 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples”. – Elemente der Mathematik 65:2 (2010).