x2y2

Hardy [1] relates the following anecdote. “I remember going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi–cab No. 1729, and remarked that the number (7 × 13 × 19) seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No, ” he replied, “it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”” Indeed, 1729 = 93 + 103 = 123 + 13. But there is another way in which this example is special. We know, since Euler, that the sum of two positive cubes is never a cube. But the above example shows that the sum of two positive cubes can do the next best thing – and that is, to miss a cube by as little as 1. Indeed, Ramanujan left for us infinitely many examples of just that phenomenon. In his so-called “Lost Notebook ” [4], he stated a result equivalent to the following