On the positive integral solutions to the Diophantine equation of n-variables (x_1+x_2+x_3+⋯+x_n )^2 = x_1 x_2 x_3…x_n

Abstract

In this paper, we discuss solvability of the Diophantine equation of the form (x1 + x2 + x3 + • • • + xn) 2 = x1x2x3• • •xn. An explicit closed form solutions are obtained by using the theory of Pell's equations which generate the different families of positive integral solutions to this equation. Some illustrative examples are inserted in order to explain the effectiveness of our results.