Let a, b, c be fixed coprime positive integers with
$\min \{a,b,c\}>1$
. We discuss the conjecture that the equation
$a^{x}+b^{y}=c^{z}$
has at most one positive integer solution
$(x,y,z)$
with
$\min \{x,y,z\}>1$
, which is far from solved. For any odd positive integer r with
$r>1$
, let
$f(r)=(-1)^{(r-1)/2}$
and
$2^{g(r)}\,\|\, r-(-1)^{(r-1)/2}$
. We prove that if one of the following conditions is satisfied, then the conjecture is true: (i)
$c=2$
; (ii) a, b and c are distinct primes; (iii)
$a=2$
and either
$f(b)\ne f(c)$
, or
$f(b)=f(c)$
and
$g(b)\ne g(c)$
.