Let $p$ be a prime number with $p>3$, $p\equiv 3\pmod{ 4}$ and let $n$ be a positive integer. In this paper, we prove that the Diophantine equation $(5pn^{ 2}-1)^{ x}+(p(p-5)n^{ 2}+1)^{ y}=(pn)^{ z}$ has only the positive integer solution $(x,y,z)=(1,1,2)$ where $pn \equiv \pm1 \pmod 5$. As an another result, we show that the Diophantine equation $(35n^{ 2}-1)^{ x}+(14n^{ 2}+1)^{ y}=(7n)^{ z}$ has only the positive integer solution $(x,y,z)=(1,1,2)$ where $n\equiv \pm 3% \pmod{ 5}$ or $5\mid n$. On the proofs, we use the properties of Jacobi symbol and Baker's method.
Categories: math.NT