Let
$a$
,
$b$
, and
$c$
be primitive Pythagorean numbers such that
${{a}^{2}}\,+\,{{b}^{2}}\,=\,{{c}^{2}}$
with
$b$
even. In this paper, we show that if
${{b}_{0}}\,\equiv \,\in \,\,\,\left( \bmod \,a \right)$
with
$\text{ }\!\!\varepsilon\!\!\text{ }\,\in \,\left\{ \pm 1 \right\}$
for certain positive divisors
${{b}_{0}}$
of
$b$
, then the Diophantine equation
${{a}^{x}}\,+\,{{b}^{y}}\,=\,{{c}^{z}}$
has only the positive solution
$\left( x,\,y,\,z \right)\,=\,\left( 2,\,2,\,2 \right)$
.