# 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

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

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 $\frac{2{u}^{2}}{u+v}$.

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 |