  We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$,
where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main
result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then
this is the only solution in integers greater than 1, with the possible
exception of finitely many values $(c,r)$. We also prove the uniqueness of such
a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we
obtain various inequalities that must be satisfied by the components of a
putative second solution.
