contraharmonic Diophantine equation
We call contraharmonic Diophantine equation the equation
(1) |
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
(2) |
For getting the general solution of the quadratic Diophantine
equation (1), one can utilise the general formulas
for
Pythagorean triples
(3) |
where the parameters , , are arbitrary positive integers with . Using (3) in (2) one obtains the result
(4) |
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
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 |