On The Diophantine Equation xa + ya = pkzb

Journal of Mathematics and Statistics

Wong, Keng Yarn¹ 

关键词: Fundamental Theorem of Arithmetic;    Congruence;    Integer Solutions;    Diophantine Equation;   

DOI  :  10.3844/jmssp.2017.38.45

学科分类：社会科学、人文和艺术（综合）

来源: Science Publications

【 摘 要 】

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.

【 授权许可】

CC BY