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 $(**)$ .

In 1956, Jeśmanowicz conjectured that, for any positive integers $m$ and $n$ with $m>n$, $\gcd (m,n)=1$ and $2\nmid m+n$, the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this paper, we prove the conjecture if $4\nmid mn$ and $y\geq 2$.