Thue's equation as a tool to solve two different problems

A Thue equation is a Diophantine equation of the form f (x, y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D-1(3)-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D-1(3)-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D-1(3)-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x(3) - y(3) = 1, we show that the only cubic-triangular number is 1.