A note on the Diophantine equations $x^2\pm 5^{\alpha }\cdot p^n=y^n$

Abstract

Suppose that $x$ is odd, $n\ge 7$ and $p\notin \{2,5\}$ are primes. In this paper, we prove that the Diophantine equations $x^2\pm 5^{\alpha }{p}^n=y^n$ have no solutions in positive integers $\alpha ,x,y$ with $gcd(x,y)=1$.

8 pages, accepted for publication in Commun. Fac. Sci. Univ. Ank. Series A1: Math. and Stat