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

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.