contraharmonic means and Pythagorean hypotenuses

One can see that all values of $c$ 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;  $9^2+{40}^2={\mathrm{\hspace{0.33em}41}}^2$.

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.  $1^{\circ }.$  Let the integer $c$ be the contraharmonic mean

$$c=\frac{u^2+v^2}{u+v}$$

of the positive integers $u$ and $v$ with  $u>v$.  Then  $u+v\mid u^2+v^2={(u+v)}^2-2uv$,  whence

$$u+v\mid \mathrm{\hspace{0.17em}2}uv,$$

and we have the positive integers

$$a=:u-v=\frac{u^2-v^2}{u+v},b=:\frac{2uv}{u+v}$$

satisfying

$$a^2+b^2=\frac{{(u^2-v^2)}^2+{(2uv)}^2}{{(u+v)}^2}=\frac{u^4-2u^2{v}^2+v^4+4u^2{v}^2}{{(u+v)}^2}=\frac{u^4+2u^2{v}^2+v^4}{{(u+v)}^2}=\frac{{(u^2+v^2)}^2}{{(u+v)}^2}=c^2.$$

$2^{\circ }.$  Suppose that $c$ is the hypotenuse of the Pythagorean triple  $(a,b,c)$,  whence  $c^2=a^2+b^2$.  Let us consider the rational numbers

$u=:{\displaystyle \frac{c+b+a}{2}},v=:{\displaystyle \frac{c+b-a}{2}}.$ (1)

If the triple is primitive (http://planetmath.org/PythagoreanTriple), then two of the integers $a,b,c$ are odd and one of them is even; if not, then similarly or all of $a,b,c$ are even.  Therefore, $c+b\pm a$ are always even and accordingly $u$ and $v$ positive integers.  We see also that  $u+v=c+b$.  Now we obtain

$u^2+v^2$ $={\displaystyle \frac{c^2+b^2+a^2+2ab+2bc+2ca+c^2+b^2+a^2-2ab+2bc-2ca}{4}}$

$={\displaystyle \frac{2c^2+2(a^2+b^2)+4bc}{4}}={\displaystyle \frac{4c^2+4bc}{4}}=c(c+b)$

$=c(u+v).$

Thus, $c$ is the contraharmonic mean ${\displaystyle \frac{u^2+v^2}{u+v}}$ of the different integers $u$ and $v$. (N.B.:  When the values of $a$ and $b$ in (1) are changed, another value of $v$ is obtained.  Cf. the Proposition 4 in the parent entry (http://planetmath.org/IntegerContraharmonicMeans).)

Title	contraharmonic means and Pythagorean hypotenuses
Canonical name	ContraharmonicMeansAndPythagoreanHypotenuses
Date of creation	2013-11-03 21:13:57
Last modified on	2013-11-03 21:13:57
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	24
Author	pahio (2872)
Entry type	Theorem
Classification	msc 11D09
Classification	msc 11D45
Classification	msc 11Z05
Classification	msc 11A05
Synonym	contraharmonic integers
Synonym	Pythagorean hypotenuses are contraharmonic means
Related topic	FirstPrimitivePythagoreanTriplets
Related topic	ProofOfPythagoreanTriplet2
Related topic	SquareOfSum
Related topic	PythagoreanTriple
Related topic	DerivationOfPythagoreanTriples
Related topic	LinearFormulasForPythagoreanTriples