contraharmonic Diophantine equation

We call contraharmonic Diophantine equation the equation

$u^2+v^2=(u+v)c$ (1)

of the three unknowns $u$, $v$, $c$ required to get only positive integer values.  The equation expresses that $c$ is the contraharmonic mean of $u$ and $v$.  As proved in the article “contraharmonic means and Pythagorean hypotenuses”, the supposition $u\ne v$ implies that the number $c$ must be the hypotenuse in a Pythagorean triple $(a,b,c)$, and if particularly , then

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

For getting the general solution of the quadratic Diophantine equation (1), one can utilise the general formulas for Pythagorean triples

$a=l\cdot (m^2-n^2),b=l\cdot 2mn,c=l\cdot (m^2+n^2)$ (3)

where the parameters $l$, $m$, $n$ are arbitrary positive integers with  $m>n$.  Using (3) in (2) one obtains the result

$\{\begin{array}{cc}{u}_1=l(m^2-mn),\hfill & \\ u_2=l(n^2+mn),\hfill & \\ v=l(m^2+mn),\hfill & \\ c=l(m^2+n^2),\hfill & \end{array}$ (4)

in which $u_1$ and $u_2$ mean the alternative values for $u$ gotten from (2) by swapping the expressions of $a$ and $b$ in (3).

It’s clear that the contraharmonic Diophantine equation has an infinite set of solutions (4).  According to the Proposition 6 of the article “integer contraharmonic means”, fixing e.g. the variable $u$ allows for the equation only a restricted number of pertinent values $v$ and $c$.  See also the alternative expressions (1) and (2) in the article “sums of two squares”.

Title	contraharmonic Diophantine equation
Canonical name	ContraharmonicDiophantineEquation
Date of creation	2013-11-19 21:49:13
Last modified on	2013-11-19 21:49:13
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Derivation
Classification	msc 11D09
Classification	msc 11D45