Sums and Differences of Three k-th Powers

If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x1 + x k 2 = N or x k 1 − x2 = N , with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N), for any ε > 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it is conjectured that the number of representations is bounded for k ≥ 4. Indeed for k ≥ 5 we know of no N for which there are two or more essentially different representations. This paper is primarily concerned with the analogous questions when one has three k-th powers. When k = 2 or 3 the equation x1 + x k 2 − x3 = N may have infinitely many solutions, as the identities