On $rac{1}{w} + rac{1}{x} + rac{1}{y} + rac{1}{z} = rac{1}{ 2} $ and some of its generalizations

摘要

In this paper, we give a straightforward approach to obtaining the solution of the Diophantine equation (rac{1}{w} + rac{1}{x} + rac{1}{y} + rac{1}{z} = rac{1}{2}). We also establish that the Diophantine equation (rac{1}{w} + rac{1}{x} + rac{1}{y} + rac{1}{z} = rac{m}{n}) for any two positive integers m and n has only a finite number of solutions in the positive integers (w, x, y), and z.