On the Diophantine equation X^{2N} + 2^{2 alpha} 5^{2 beta} p^{2 gamma} = Z^5

TL;DR

This paper proves the non-existence of solutions for a specific class of Diophantine equations involving powers and prime factors, under certain conditions on the variables and parameters.

Contribution

It establishes a new non-existence result for a family of Diophantine equations with mixed power and prime factor conditions.

Findings

No solutions exist for the given equation under specified conditions.

The proof applies number theoretic methods to demonstrate non-solvability.

Results extend understanding of power equations with prime factor constraints.

Abstract

We prove that for each odd prime p, positive integer alpha, and non-negative integers beta and gamma, the Diophantine equation X^{2N} + 2^{2 alpha} 5^{2 beta} p^{2 gamma} = Z^5 has no solution with X, Z, N in Z^+, N > 1, and gcd(X,Z) = 1.