integer harmonic means
of and 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
. As a matter of fact, one
cathetus of a right triangle is the harmonic mean of the same
positive integers and 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 , , , when .
Some of the propositions concerning the integer contraharmonic means directly imply corresponding propositions of the integer harmonic means:
Proposition 1. For any value of , there are at least two greater values
of such that in (2) is an integer.
Proposition 2. For all , a necessary condition for is that
Proposition 3. If is an odd prime number, then the values (3) are the only possibilities for enabling integer harmonic means with .
Proposition 5. When the harmonic mean of two different positive integers and is an integer, their sum is never squarefree.
Proposition 6. For each integer there are only a finite number of solutions of the Diophantine equation (2).
Proposition 6 follows also from the inequality
which yields the estimation
(cf. the above table). This is of course true for any harmonic means of positive numbers and . The difference of and is .
The estimation (4) implies that the number of solutions is less than . 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 .
|Title||integer harmonic means|
|Date of creation||2013-11-06 17:18:49|
|Last modified on||2013-11-06 17:18:49|
|Last modified by||pahio (2872)|