ON WARING’S PROBLEM IN SUMS OF THREE CUBES FOR SMALLER POWERS

On Powers as Sums of Two Cubes

In a paper of Kraus, it is proved that x 3 + y 3 = z p for p 17 has only trivial primitive solutions, provided that p satisses a relatively mild and easily tested condition. In this article we prove that the primitive solutions of x 3 + y 3 = z p with p = 4; 5; 7; 11; 13, correspond to rational points on hyperelliptic curves with Jaco-bians of relatively small rank. Consequently, Chabauty metho...

Sums of Squares, Cubes, and Higher Powers

On the Waring–goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

(1.1) p1 + · · ·+ ps = n, where p1, . . . , ps are prime unknowns. It is conjectured that for any pair of integers k, s ∈ N with s ≥ k + 1 there exist a fixed modulus qk,s and a collection Nk,s of congruence classes mod qk,s such that (1.1) is solvable for all sufficiently large n ∈ Nk,s. While a proof of this conjecture appears to be beyond the reach of present methods, some significant progre...

New sums of three cubes

Introduction. It is a long standing problem whether every rational integer n 6≡ 4, 5 (mod 9) may be written as a sum of three integral cubes. According to the web page http://cr.yp.to/threecubes.html of Daniel Bernstein, the first attacks by computer were carried out as early as in 1955. Nevertheless, for example for n = 3, there is still no solution known different from the obvious (1, 1, 1), ...

Correlation Estimates for Sums of Three Cubes

We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of systems of homogeneous linear equations in sums of three positive cubes whenever these systems have more than twice as many variables as equations.