contraharmonic means and Pythagorean hypotenuses
One can see that all values of in the table of the parent entry (http://planetmath.org/IntegerContraharmonicMeans) are hypotenuses in a right triangle with integer sides (http://planetmath.org/Triangle). E.g., 41 is the contraharmonic mean of 5 and 45; .
Theorem. Any integer contraharmonic mean of two different positive integers is the hypotenuse of a Pythagorean triple. Conversely, any hypotenuse of a Pythagorean triple is contraharmonic mean of two different positive integers.
Proof. Let the integer be the contraharmonic mean
of the positive integers and with . Then , whence
and we have the positive integers
Suppose that is the hypotenuse of the Pythagorean triple , whence . Let us consider the rational numbers
If the triple is primitive (http://planetmath.org/PythagoreanTriple), then two of the integers are odd and one of them is even; if not, then similarly or all of are even. Therefore, are always even and accordingly and positive integers. We see also that . Now we obtain
Thus, is the contraharmonic mean of the different integers and . (N.B.: When the values of and in (1) are changed, another value of is obtained. Cf. the Proposition 4 in the parent entry (http://planetmath.org/IntegerContraharmonicMeans).)
- 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples”. – Elemente der Mathematik 65:2 (2010).
|Title||contraharmonic means and Pythagorean hypotenuses|
|Date of creation||2013-11-03 21:13:57|
|Last modified on||2013-11-03 21:13:57|
|Last modified by||pahio (2872)|
|Synonym||Pythagorean hypotenuses are contraharmonic means|