contraharmonic Diophantine equation
We call contraharmonic Diophantine equation the equation
of the three unknowns , , required to get only positive integer values. The equation expresses that is the contraharmonic mean of and . As proved in the article “contraharmonic means and Pythagorean hypotenuses”, the supposition implies that the number must be the hypotenuse in a Pythagorean triple , and if particularly , then
where the parameters , , are arbitrary positive integers with . Using (3) in (2) one obtains the result
in which and mean the alternative values for gotten from (2) by swapping the expressions of and in (3).
It’s clear that the contraharmonic Diophantine equation has an
infinite set of solutions (4). According to the Proposition
6 of the article “integer contraharmonic means”, fixing e.g.
the variable allows for the equation only a restricted
number of pertinent values and . See also the
alternative expressions (1) and (2) in the article “sums of
|Title||contraharmonic Diophantine equation|
|Date of creation||2013-11-19 21:49:13|
|Last modified on||2013-11-19 21:49:13|
|Last modified by||pahio (2872)|