In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A,B,C)$ of the Diophantine equation $-(2^p·A^6)+B^3=C^2$ for any positive integral values of $p$ when $p\equiv 1$ (mod 6) or $p\equiv 2$ (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.