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Let $a$, $b$, $c$ be distinct primes with $a<b$. Let $S(a,b,c)$ denote thenumber 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 atriple of distinct primes for which $S(a,b,c)>1$ and $(a,b,c)$ is not one ofthe six known such triples then $(a,b,c)$ must be one of three cases. In thepresent paper, we eliminate two of these cases (using the special properties ofcertain continued fractions for one of these cases, and using a result ofDirichlet on quartic residues for the other). Then we show that the singleremaining 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 leastone 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 bean octic residue modulo $c$ except for one specific case; $2 \mid v_2(b-1) \lev_2(c-1)$ (where $v_2(n)$ satisfies $2^{v_2(n)} \parallel n$); there must beexactly 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 aconjecture put forward in \cite{ScSt6} and improve results in \cite{LeSt}.

On a conjecture concerning the number of solutions to $a^x+b^y=c^z$, II