# A note on the Diophantine equations $x^{2}\pm5^{\alpha}\cdot p^{n}=y^{n}$

## TL;DR

This paper proves that certain exponential Diophantine equations involving powers of 5, a prime p not equal to 2 or 5, and exponents n ≥ 7 have no solutions under specified conditions, extending understanding of these equations.

## Contribution

It establishes the non-existence of solutions for the equations x^2 ± 5^α p^n = y^n with specified constraints, for primes p ≠ 2,5 and n ≥ 7.

## Key findings

- No solutions in positive integers for the given equations under specified conditions.
- Extends previous results on exponential Diophantine equations involving prime powers.
- Provides new insights into the structure of solutions for equations with mixed powers.

## Abstract

Suppose that $x$ is odd, $n\geq7$ and $p\notin\{2,5\}$ are primes. In this paper, we prove that the Diophantine equations $x^{2}\pm5^{\alpha}p^{n}=y^{n}$ have no solutions in positive integers $\alpha,x,y$ with $gcd(x,y)=1$.