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
$${a}^{2}+{b}^{2}=\frac{{({u}^{2}-{v}^{2})}^{2}+{(2uv)}^{2}}{{(u+v)}^{2}}=\frac{{u}^{4}-2{u}^{2}{v}^{2}+{v}^{4}+4{u}^{2}{v}^{2}}{{(u+v)}^{2}}=\frac{{u}^{4}+2{u}^{2}{v}^{2}+{v}^{4}}{{(u+v)}^{2}}=\frac{{({u}^{2}+{v}^{2})}^{2}}{{(u+v)}^{2}}={c}^{2}.$$ |
${2}^{\circ}.$ 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
${u}^{2}+{v}^{2}$ | $={\displaystyle \frac{{c}^{2}+{b}^{2}+{a}^{2}+2ab+2bc+2ca+{c}^{2}+{b}^{2}+{a}^{2}-2ab+2bc-2ca}{4}}$ | ||
$={\displaystyle \frac{2{c}^{2}+2({a}^{2}+{b}^{2})+4bc}{4}}={\displaystyle \frac{4{c}^{2}+4bc}{4}}=c(c+b)$ | |||
$=c(u+v).$ |
Thus, $c$ is the contraharmonic mean $\frac{{u}^{2}+{v}^{2}}{u+v}$ 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).)
References
- 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples”. – Elemente der Mathematik 65:2 (2010).
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 |