An Interesting Diophantine Problem

This paper concerns with an interesting Diophantine problem and aims at determining explicitly three distinct non-zero integers a, b, c such that the sum of any pair of them is a perfect square and twice the sum of the three integers is a perfect cube. Different methods have been considered to obtain the three required integers a, b, c. This shows that there are many triples in integers, each satisfying the considered kind of pattern among its members.