VanderBurgh (2010) and Korsky (2026) use harmonic triple for a triple 
of positive integers
satisfying 
and
 |
(1)
|
In particular, such a triple is harmonic if the reciprocals of its terms form an arithmetic sequence
with common difference 
where
 |
(2)
|
There is a one-to-one correspondence between the set of equivalence classes of harmonic
triples and the set of equivalence classes of geometric
triples. Here, two triples 
and 
are said to be equivalent
if 
,
or equivalently if there exists a positive real
number 
such that 
.
Shirali (2013a, 2013b) use harmonic triple for a triple 
of positive integers satisfying
 |
(3)
|
This is a different diophantine equation from the reciprocal-arithmetic-progression condition above. All its positive integer
solutions have the form 
for positive integers 
, 
, and 
with 
and 
relatively prime. In this
usage, a harmonic triple is called primitive when the greatest
common divisor 
is 1, equivalently when 
.
