Integer Solutions to Non-Homogeneous Cubic Equation with Two Unknowns \(a (x - y)^3= 8 b x y ; a, b \in z - \) {0}

Abstract

Diophantine equations, one of the interesting areas in Number theory, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. This chapter discusses on finding many solutions in integers to the cubic equation with two unknowns given by \(a (x-y)^3 = 8b x y\) ; \(a,b \in z -\) {0}, as the cubic equations fall into the theory of elliptic curves. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers are discussed in properties. These special numbers are unique and have attractive characterization that sets them apart from other numbers. The process of formulating second-order Ramanujan numbers with base numbers as real integers is illustrated through examples.  The process for getting a sequence of Diophantine triples with suitable properties and Dio-3 tuples with suitable properties is illustrated.