On the Diophantine equation $(5p{n}^2-1{)}^x+(p(p-5)n^2+1{)}^y=(pn{)}^z$

Abstract

Let $p$ be a prime number with $p>3$, $p\equiv 3(\mathrm{mod}4)$ and let $n$ be a positive integer. In this paper, we prove that the Diophantine equation $(5p{n}^2-1{)}^x+(p(p-5)n^2+1{)}^y=(pn{)}^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$ where $pn\equiv \pm 1(\mathrm{mod}5)$. 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 $n\equiv \pm 3$ or $5\mid n$. On the proofs, we use the properties of Jacobi symbol and Baker's method.

9 pages