If
a
,
b
a, b
and
n
n
are positive integers with
b
≥
a
b \geq a
and
n
≥
3
n \geq 3
, then the equation of the title possesses at most one solution in positive integers
x
x
and
y
y
, with the possible exceptions of
(
a
,
b
,
n
)
( a, b, n )
satisfying
b
=
a
+
1
b = a + 1
,
2
≤
a
≤
min
{
0.3
n
,
83
}
2 \leq a \leq \min \{ 0.3 n, 83 \}
and
17
≤
n
≤
347
17 \leq n \leq 347
. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions, the theory of linear forms in logarithms and recent computational methods related to lattice-basis reduction. Additionally, we compare and contrast a number of these last mentioned techniques.