contraharmonic means and Pythagorean hypotenuses

One can see that all values of c in the table of the parent entry ( are hypotenusesMathworldPlanetmath in a right triangleMathworldPlanetmath with integer sides (  E.g., 41 is the contraharmonic mean of 5 and 45;  92+402= 412.

TheoremMathworldPlanetmath.  Any integer contraharmonic mean of two different positive integers is the hypotenuse of a Pythagorean tripleMathworldPlanetmath.  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


of the positive integers u and v with  u>v.  Then  u+vu2+v2=(u+v)2-2uv,  whence

u+v 2uv,

and we have the positive integers




2.  Suppose that c is the hypotenuse of the Pythagorean triple  (a,b,c),  whence  c2=a2+b2.  Let us consider the rational numbersPlanetmathPlanetmathPlanetmath

u=:c+b+a2,v=:c+b-a2. (1)

If the triple is primitive (, 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

u2+v2 =c2+b2+a2+2ab+2bc+2ca+c2+b2+a2-2ab+2bc-2ca4

Thus, c is the contraharmonic mean u2+v2u+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 PropositionPlanetmathPlanetmath 4 in the parent entry (


  • 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