Journal of Mathematics and Statistics

On The Diophantine Equation xa + ya = pkzb

Keng Yarn Wong and Hailiza Kamarulhaili

DOI : 10.3844/jmssp.2017.38.45

Journal of Mathematics and Statistics

Volume 13, Issue 1

Pages 38-45


In this study, we consider the Diophantine equation xa + ya = pkzb 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 xa + ya = pkzb 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.


© 2017 Keng Yarn Wong and Hailiza Kamarulhaili. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.