contraharmonic means and Pythagorean hypotenuses

contraharmonic means and Pythagorean hypotenuses

One can see that all values of

c

in the table of the parent entry (

http://

planetmath

.org/IntegerContraharmonicMeans

) are

hypotenuses

in a

right triangle

with

integer

sides (

http://planetmath.org/

Triangle

).  E.g., 41 is the

contraharmonic mean

of 5 and 45;

9

2

+

40

2

=

41

2

.

Theorem

.

Any

integer contraharmonic mean

of two different

positive integers

is the hypotenuse of a

Pythagorean triple

.

Conversely

, any hypotenuse of a Pythagorean triple is contraharmonic mean of two different positive integers.

Proof.

1

∘

.

Let the integer

c

be the contraharmonic mean

c

=

u

2

+

v

2

u

+

v

of the positive integers

u

and

v

with

u

>

v

.  Then

u

+

v

∣

u

2

+

v

2

=

(

u

+

v

)

2

-

2

u

v

,
whence

u

+

v

∣

2

u

v

,

and we have the positive integers

a

=

:

u

-

v

=

u

2

-

v

2

u

+

v

,

b

=

:

2

⁢

u

⁢

v

u

+

v

satisfying

a

2

+

b

2

=

(

u

2

-

v

2

)

2

+

(

2

⁢

u

⁢

v

)

2

(

u

+

v

)

2

=

u

4

-

2

⁢

u

2

⁢

v

2

+

v

4

+

4

⁢

u

2

⁢

v

2

(

u

+

v

)

2

=

u

4

+

2

⁢

u

2

⁢

v

2

+

v

4

(

u

+

v

)

2

=

(

u

2

+

v

2

)

2

(

u

+

v

)

2

=

c

2

.

2

∘

.

Suppose that

c

is the hypotenuse of the Pythagorean triple

(

a

,

b

,

c

)

,  whence

c

2

=

a

2

+

b

2

.  Let us consider the

rational numbers

u

=

:

c

+

b

+

a

2

,

v

=

:

c

+

b

-

a

2

.

(1)

If the triple is

primitive

(

http://planetmath.org/PythagoreanTriple

), then two of the integers

a

,

b

,

c

are odd and one of them is even; if not, then similarly or all of

a

,

b

,

c

are even.  Therefore,

c

+

b

±

a

are always even and accordingly

u

and

v

positive integers.  We see also that

u

+

v

=

c

+

b

.  Now we obtain

u

2

+

v

2

=

c

2

+

b

2

+

a

2

+

2

⁢

a

⁢

b

+

2

⁢

b

⁢

c

+

2

⁢

c

⁢

a

+

c

2

+

b

2

+

a

2

-

2

⁢

a

⁢

b

+

2

⁢

b

⁢

c

-

2

⁢

c

⁢

a

4

=

2

⁢

c

2

+

2

⁢

(

a

2

+

b

2

)

+

4

⁢

b

⁢

c

4

=

4

⁢

c

2

+

4

⁢

b

⁢

c

4

=

c

⁢

(

c

+

b

)

=

c

⁢

(

u

+

v

)

.

Thus,

c

is the contraharmonic mean

u

2

+

v

2

u

+

v

of the different integers

u

and

v

. (N.B.:  When the values of

a

and

b

in (1) are changed, another value of

v

is obtained.  Cf. the

Proposition

4 in the
parent entry (

http://planetmath.org/IntegerContraharmonicMeans

).)

References

1

J. Pahikkala

: “On contraharmonic mean and Pythagorean triples”.  –

Elemente der Mathematik

65

:2 (2010).

Title

contraharmonic means and Pythagorean hypotenuses

Canonical

name

ContraharmonicMeansAndPythagoreanHypotenuses

Date of creation

2013-11-03 21:13:57

Last modified on

2013-11-03 21:13:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

24

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11D09

Classification

msc 11D45

Classification

msc 11Z05

Classification

msc 11A05

Synonym

contraharmonic integers

Synonym

Pythagorean hypotenuses are contraharmonic means

Related topic

FirstPrimitivePythagoreanTriplets

Related topic

ProofOfPythagoreanTriplet2

Related topic

SquareOfSum

Related topic

PythagoreanTriple

Related topic

DerivationOfPythagoreanTriples

Related topic

LinearFormulasForPythagoreanTriples

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