On The Diophantine Equation x^a + y^a = p^kz^b

Abstract

In this study, we consider the Diophantine equation x^a + y^a = p^kz^b where p is a prime number, gcd(a, b) = 1 and k,a,b∈Z⁺. We solve this equation parametrically by considering different cases of x and y and find that there exist infinitely many nontrivial integer solutions, where the formulated parametric solutions solve x^a + y^a = p^kz^b completely for the case of x = y, x = −y, and either x or y is zero (not both zero). For the case of |x| ≠ |y| and both x and y nonzero, not every solution (x,y,z) is in the parametric forms proposed in Theorem 5, although any (x,y,z) in these parametric forms solves the Diophantine equation.