Let
be a primitive Pythagorean triple satisfying
In 1956, Jesmanowicz conjectured that the only positive integer solution of the exponential Diophantine equation
is
For the primitive Pythagorean triple
with
positive integers such that gcd
,
, many special cases of Jesmanowicz' conjecture have been settled under the condition that
. In this paper, using the theory of quartic residue character and elementary method together with some of Miyazaki's results, we show that Jesmanowicz' conjecture is true if
or
.