Jeśmanowicz conjectured that
$(x,y,z)=(2,2,2)$
is the only positive integer solution of the equation
$(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$
, where n is a positive integer and f, g are positive integers such that
$f>g$
,
$\gcd (\kern1.5pt f,g)=1$
and
$f \not \equiv g\pmod 2$
. Using Baker’s method, we prove that: (i) if
$n>1$
,
$f \ge 98$
and
$g=1$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$x>z>y$
; and (ii) if
$n>1$
,
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)\; 2 \nmid s$
and
$s<2^{r/2}$
, then
$(*)$
has no positive integer solutions
$(x,y,z)$
with
$y>z>x$
. Thus, Jeśmanowicz’ conjecture is true if
$f=2^rs^2$
and
$g=1$
, where r, s are positive integers satisfying
$(**)$
.