  Let $a$, $b$, $c$ be distinct primes with $a<b$. Let $S(a,b,c)$ denote the
number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y =
c^z$. In a previous paper \cite{LeSt} it was shown that if $(a,b,c)$ is a
triple of distinct primes for which $S(a,b,c)>1$ and $(a,b,c)$ is not one of
the six known such triples then $(a,b,c)$ must be one of three cases. In the
present paper, we eliminate two of these cases (using the special properties of
certain continued fractions for one of these cases, and using a result of
Dirichlet on quartic residues for the other). Then we show that the single
remaining case requires severe restrictions, including the following: $a=2$, $b
\equiv 1 \bmod 48$, $c \equiv 17 \mod 48$, $b > 10^9$, $c > 10^{18}$; at least
one of the multiplicative orders $u_c(b)$ or $u_b(c)$ must be odd (where
$u_p(n)$ is the least integer $t$ such that $n^t \equiv 1 \bmod p$); 2 must be
an octic residue modulo $c$ except for one specific case; $2 \mid v_2(b-1) \le
v_2(c-1)$ (where $v_2(n)$ satisfies $2^{v_2(n)} \parallel n$); there must be
exactly two solutions $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ with $1 = z_1 <
z_2$ and either $x_1 \ge 28$ or $x_2 \ge 88$. These results support a
conjecture put forward in \cite{ScSt6} and improve results in \cite{LeSt}.
