The representation of numbers as sums of unlike powers. II

Abstract

n= Zx,+ i=1 The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wol], [VW2]). In section 2, an algorithm developed in [Fo] for optimizing the parameters in mixed power mean value theorems is generalized and analyzed. Section 3 details a more sophisticated method of generating mixed power mean value theorems, by a limited adaptation of the iterative method itself. The form of these estimates offers many advantages over those of section 2, and provides the key to the elimination of the 16th power from (1.1). These mean value theorems are then applied to the proof of Theorem 1 in section 4. The tools developed here are applicable to a wide range of mixed power representation problems, and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n, when the lowest power used is a kth power instead of a square. Throughout, n is a large natural number whose representation as a sum of mixed powers is at issue, and E is an arbitrarily small positive real number. Constants implied by the Landau Oand Vinogradov <<-symbols may depend on E or k. For a real number x, write e(x) for e27iX, [x] for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer. Unless otherwise specified, lowercase Latin letters denote natural numbers and Greek letters denote real numbers. Let A(P, R) denote the set of natural numbers not exceeding P with no