integer harmonic means

Let $u$ and $v$ be positive integers.  As is seen in the parent entry (http://planetmath.org/IntegerContraharmonicMeans), there exist nontrivial cases ($u\ne v$) where their contraharmonic mean

$c:={\displaystyle \frac{u^2+v^2}{u+v}}=u+v-{\displaystyle \frac{2uv}{u+v}}$ (1)

is an integer.  Because the subtrahend of the last is the harmonic mean of $u$ and $v$, the equation means that the contraharmonic mean $c$ and the harmonic mean

$h:={\displaystyle \frac{2uv}{u+v}}$ (2)

of $u$ and $v$ are simultaneously integers.

The integer contraharmonic mean of two distinct positive integers ranges exactly the set of hypotenuses of Pythagorean triples (see contraharmonic integers), but the integer harmonic mean of two distinct positive integers the wider set  $\{3,\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}5},\mathrm{\hspace{0.17em}6},\mathrm{\dots }\}$.  As a matter of fact, one cathetus of a right triangle is the harmonic mean of the same positive integers $u$ and $v$ the contraharmonic mean of which is the hypotenuse of the triangle (see Pythagorean triangle (http://planetmath.org/PythagoreanTriangle)).

The following table allows to compare the values of $u$, $v$, $c$, $h$ when  .

$u$	$2$	$3$	$3$	$4$	$4$	$5$	$5$	$6$	$6$	$6$	$6$	$7$	$7$	$8$	$8$	$8$	$9$	$9$	$\mathrm{\dots }$
$v$	$6$	$6$	$15$	$12$	$28$	$20$	$45$	$12$	$18$	$30$	$66$	$42$	$91$	$24$	$56$	$120$	$18$	$45$	$\mathrm{\dots }$
$c$	$5$	$5$	$13$	$10$	$25$	$17$	$41$	$10$	$15$	$26$	$61$	$37$	$85$	$20$	$50$	$113$	$15$	$39$	$\mathrm{\dots }$
$h$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$8$	$9$	$10$	$11$	$12$	$13$	$12$	$14$	$15$	$12$	$15$	$\mathrm{\dots }$

Some of the propositions concerning the integer contraharmonic means directly imply corresponding propositions of the integer harmonic means:

Proposition 1.  For any value of $u>2$, there are at least two greater values

$v_1:=(u-1)u,v_2:=(2u-1)u$ (3)

of $v$ such that $h$ in (2) is an integer.

Proposition 2.  For all  $u>1$, a necessary condition for $h\in \mathbb{Z}$  is that

$$\gcd (u,v)>\mathrm{\hspace{0.33em}1}.$$

Proposition 3.  If $u$ is an odd prime number, then the values (3) are the only possibilities for  $v>u$  enabling integer harmonic means with $u$.

Proposition 5.  When the harmonic mean of two different positive integers $u$ and $v$ is an integer, their sum is never squarefree.

Proposition 6.  For each integer $u>0$ there are only a finite number of solutions  $(u,v,h)$  of the Diophantine equation (2).

Proposition 6 follows also from the inequality

$$\frac{1}{h}=\frac{1}{2}\left(\frac{1}{u}+\frac{1}{v}\right)>\frac{1}{2u}$$

which yields the estimation

(4)

(cf. the above table).  This is of course true for any harmonic means $h$ of positive numbers $u$ and $v$.  The difference of $2u$ and $h$ is $\frac{2u^2}{u+v}$.

The estimation (4) implies that the number of solutions is less than $2u$.  From the proof of the corresponding proposition in the http://planetmath.org/node/11241parent entry one can see that the number in fact does not exceed $u-1$.

Title	integer harmonic means
Canonical name	IntegerHarmonicMeans
Date of creation	2013-11-06 17:18:49
Last modified on	2013-11-06 17:18:49
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	20
Author	pahio (2872)
Entry type	Topic
Classification	msc 11Z05
Classification	msc 11D45
Classification	msc 11D09
Classification	msc 11A05
Related topic	HarmonicMean
Related topic	HarmonicMeanInTrapezoid