The term harmonic triple is used by VanderBurgh (2010) and Korsky (2026) 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 progression
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 
belong to the same equivalence
class iff 
, or equivalently if there exists a positive real number 
such that 
.
Shirali (2013ab) uses 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 
.
