LAUSR

A positive integer, which can be written as the sum of two positive cubes in two different ways, is known as a “Ramanujan number”. The most famous example is 1729=103+93=123+13, which was identified by Ramanujan as the lowest such number. In this paper, we consider the homogeneous cubic Diophantine equation x3+y3=u3+v3, where there is no restriction on the signs of the integers x,y,u,v. We show that every solution can be written in terms of two parameters in the ring Z−3. It is also shown that solutions with arbitrarily high values of max(|x|,|y|,|u|,|v|) arise amongst the primitive solutions.