# 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 |