Let a, b, c be fixed coprime positive integers with
$\min \{ a,b,c \}>1$
. Let
$N(a,b,c)$
denote the number of positive integer solutions
$(x,y,z)$
of the equation
$a^x + b^y = c^z$
. We show that if
$(a,b,c)$
is a triple of distinct primes for which
$N(a,b,c)>1$
and
$(a,b,c)$
is not one of the six known such triples, then
$c>10^{18}$
, and there are exactly two solutions
$(x_1, y_1, z_1)$
,
$(x_2, y_2, z_2)$
with
$2 \mid x_1$
,
$2 \mid y_1$
,
$z_1=1$
,
$2 \nmid y_2$
,
$z_2>1$
, and, taking
$a<b$
, we must have
$a=2$
,
$b \equiv 1 \bmod 12$
,
$c \equiv 5\, \mod 12$
, with
$(a,b,c)$
satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to
$a^x + b^y = c^z$
’, Publ. Math. Debrecen88 (2016), 131–138].