Non-homogeneous binary cubic equation a(x − y)3 = 8 x y , a > 0

Abstract

Asearch is made for determining the different solutions in integers to the binary cubic equation a(x − y)3 = 8xy,a > 0 by employing linear transformations. Some special relations for the solutions are obtained. The process for getting second order Ramanujan numbers, sequence of Diophantine triples with suitable property and Dio-3 tuples is also illustrated.