integer harmonic means

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

≠

v

) where their

contraharmonic mean

c

:=

u

2

+

v

2

u

+

v

=

u

+

v

-

2

⁢

u

⁢

v

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

:=

2

⁢

u

⁢

v

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

,

…

}

.  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

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

v

1

:=

(

u

-

1

)

⁢

u

,

v

2

:=

(

2

⁢

u

-

1

)

⁢

u

(3)

of

v

such that

h

in (2) is an integer.

Proposition 2.

For all

u

>

1

, a necessary condition for

h

∈

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

1

h

=

1

2

⁢

(

1

u

+

1

v

)

>

1

2

⁢

u

which yields the estimation

0

<

h

<

2

⁢

u

(4)

(cf. the above table).  This is of course true for any harmonic means

h

of positive numbers

u

and

v

.  The difference of

2

⁢

u

and

h

is

2

⁢

u

2

u

+

v

.

The estimation (4) implies that the number of solutions is less than

2

⁢

u

.  From the proof of the corresponding proposition in the

http://planetmath.org/node/11241

parent 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

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