Research Article Open Access

On The Diophantine Equation xa + ya = pkzb

Keng Yarn Wong1 and Hailiza Kamarulhaili1
  • 1 Universiti Sains Malaysia, Malaysia

Abstract

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.

Journal of Mathematics and Statistics
Volume 13 No. 1, 2017, 38-45

DOI: https://doi.org/10.3844/jmssp.2017.38.45

Submitted On: 3 February 2017 Published On: 11 March 2017

How to Cite: Wong, K. Y. & Kamarulhaili, H. (2017). On The Diophantine Equation xa + ya = pkzb. Journal of Mathematics and Statistics, 13(1), 38-45. https://doi.org/10.3844/jmssp.2017.38.45

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Keywords

  • Diophantine Equation
  • Integer Solutions
  • Congruence
  • Fundamental Theorem of Arithmetic