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Let $u$ and $v$ be positive^{} integers. As is seen in the parent entry, there exist nontrivial cases
($u\neq v$) where their contraharmonic mean

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

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

$\displaystyle h\;:=\;\frac{2uv}{u\!+\!v}$ | (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
$\{3,\,4,\,5,\,6,\,\ldots\}$. 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).

The following table allows to compare the values of $u$, $v$, $c$, $h$ when $1\,<\,u\,<\,v$.

$u$ | $2$ | $3$ | $3$ | $4$ | $4$ | $5$ | $5$ | $6$ | $6$ | $6$ | $6$ | $7$ | $7$ | $8$ | $8$ | $8$ | $9$ | $9$ | $...$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$v$ | $6$ | $6$ | $15$ | $12$ | $28$ | $20$ | $45$ | $12$ | $18$ | $30$ | $66$ | $42$ | $91$ | $24$ | $56$ | $120$ | $18$ | $45$ | $...$ |

$c$ | $5$ | $5$ | $13$ | $10$ | $25$ | $17$ | $41$ | $10$ | $15$ | $26$ | $61$ | $37$ | $85$ | $20$ | $50$ | $113$ | $15$ | $39$ | $...$ |

$h$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $12$ | $14$ | $15$ | $12$ | $15$ | $...$ |

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

$\displaystyle v_{1}\;:=\;(u\!-\!1)u,\quad v_{2}\;:=\;(2u\!-\!1)u$ | (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

$\gcd(u,\,v)\;>\;1.$ |

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

$\displaystyle 0\;<\;h\;<\;2u$ | (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 parent entry one can see that the number in fact does not exceed $u\!-\!1$.

## Mathematics Subject Classification

11Z05*no label found*11D45

*no label found*11D09

*no label found*11A05

*no label found*

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