contraharmonic Diophantine equation

We call contraharmonic Diophantine equation the equation

$\displaystyle u^{2}+v^{2}\;=\;(u+v)c$

(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 $u\neq v$ implies that the number $c$ must be the hypotenuse in a Pythagorean triple $(a,b,c)$ , and if particularly $u<v$ , then

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

(2)

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

$\displaystyle a\;=\;l\!\cdot\!(m^{2}-n^{2}),\quad b\;=\;l\!\cdot\!2mn,\quad c% \;=\;l\!\cdot\!(m^{2}+n^{2})$

(3)

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

$\displaystyle\begin{cases}u_{1}\;=\;l(m^{2}-mn),\\ u_{2}\;=\;l(n^{2}+mn),\\ v\;=\;l(m^{2}+mn),\\ c\;=\;l(m^{2}+n^{2}),\end{cases}$

(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