Let \(\nu(x)\) be the number of positive integers \(n\leq x\) which have two or more distinct representations as a sum of two cubes of non-negative integers. It was shown by C. Hooley [J. Reine Angew. Math. 314, 146-173 (1980; Zbl 0423.10026)] that \(\nu(x)\ll x^{5/9+ \varepsilon}\) for any \(\varepsilon>0\). Hooley’s proof used a sieve method coupled with estimates for exponential sums derived from Deligne’s work. The present paper gives a completely elementary proof of Hooley’s result.
The problem can be reformulated by asking about nontrivial integer points on the cubic surface \(X^ 3_ 1+ X^ 3_ 2+ X^ 3_ 3+ X^ 3_ 4 =0\), lying in the box \(| X_ i |\leq P\). Here a point is regarded as trivial if it lies on a rational line in the surface. One can show, by simple adaptions of either Hooley’s approach or Wooley’s, that the number of nontrivial points above is \(O( P^{5/3+ \varepsilon})\).
After studying the present paper the reviewer realized that he had established a more general result in 1992, using a similar method to Wooley’s, namely that the above bound for nontrivial points holds for any cubic surface with 3 coplanar rational lines. Unfortunately he had failed to realize that the rational lines \(x_ 1+ x_ 2= x_ 3+ x_ 4=0\), \(x_ 1+ x_ 3= x_ 2+ x_ 4 =0\) and \(x_ 1+ x_ 4= x_ 2+ x_ 3 =0\) in the diagonal surface above, all lie in the plane \(x_ 1+ x_ 2+ x_ 3+ x_ 4 =0!\).