Let
$a,b,c$
be a primitive Pythagorean triple and set
$a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$
, where
$m$
and
$n$
are positive integers with
$m>n$
,
$\text{gcd}(m,n)=1$
and
$m\not \equiv n~(\text{mod}~2)$
. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation
$(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$
is
$(x,y,z)=(2,2,2)$
. We use biquadratic character theory to investigate the case with
$(m,n)\equiv (2,3)~(\text{mod}~4)$
. We show that Jeśmanowicz’ conjecture is true in this case if
$m+n\not \equiv 1~(\text{mod}~16)$
or
$y>1$
. Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if
$(m,n)\equiv (2,3)~(\text{mod}~4)$
and
$n<100$
.