contraharmonic Diophantine equation

We call contraharmonic Diophantine equation the equation

u2+v2=(u+v)c (1)

of the three unknowns u, v, c required to get only positive integer values.  The equation expresses that c is the contraharmonic mean of u and v.  As proved in the article “contraharmonic means and Pythagorean hypotenuses”, the supposition uv implies that the number c must be the hypotenuseMathworldPlanetmath in a Pythagorean tripleMathworldPlanetmath (a,b,c), and if particularly u<v, then

u=c+b-a2,v=c+b+a2. (2)

For getting the general solution of the quadratic Diophantine equationMathworldPlanetmath (1), one can utilise the general formulasMathworldPlanetmathPlanetmath for Pythagorean triples

a=l(m2-n2),b=l2mn,c=l(m2+n2) (3)

where the parameters l, m, n are arbitrary positive integers with  m>n.  Using (3) in (2) one obtains the result

{u1=l(m2-mn),u2=l(n2+mn),v=l(m2+mn),c=l(m2+n2), (4)

in which u1 and u2 mean the alternative values for u gotten from (2) by swapping the expressions of a and b in (3).

It’s clear that the contraharmonic Diophantine equation has an infinite setMathworldPlanetmath of solutions (4).  According to the PropositionPlanetmathPlanetmath 6 of the article “integer contraharmonic means”, fixing e.g. the variable u allows for the equation only a restricted number of pertinent values v and c.  See also the alternative expressions (1) and (2) in the article “sums of two squares”.

Title contraharmonic Diophantine equation
Canonical name ContraharmonicDiophantineEquation
Date of creation 2013-11-19 21:49:13
Last modified on 2013-11-19 21:49:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Derivation
Classification msc 11D09
Classification msc 11D45