On the diophantine equation ∏³_i=1x_i — n³ = m ∏³_i=1(x_i — n); m and n are positive integers

The Diophantine equation under consideration is a cubic non-linear Diophantine equation. We have to obtain its positive integral solutions for different values of m and n. For n = 1, let x₁ = a, x₂ = band x₃ = c. Then abc — 1 = m(a — 1)(b — 1)(c — 1). It means we have to find the values of a, b and c such that abc — 1 is divisible by (a — 1 )(b — 1 )(c — 1) where 1 < a < b < c. This problem was posed initially in April 2017 in math.stakexchange.com and by American University, Sharjah in March 2021. The solutions obtained are (a,b,c)= (2,4,8) and (3,5,15). In this paper, the problem has been discussed for other values of n.