Koichi Kawada

Additive representation in thin sequences, I: Waring's problem for cubes

Abstract In this paper we investigate representation of numbers from certain thin sequences like the squares or cubes by sums of cubes. It is shown, in particular, that almost all values of an integral cubic polynomial are sums of six cubes. The methods are very flexible and may be applied to many related problems.

A Ternary Problem in Additive Prime Number Theory

We discuss the solubility of the ternary equations x 2+y 3+z k = n for an integer k with 3 ≤ k ≤ 5 and large integers n, where two of the variables are primes, and the remaining one is an almost prime. We are also concerned with related quaternary problems. As usual, an integer with at most r prime factors is called a P r -number. We shall show, amongst other things, that for almost all odd n, the equation x 2 +p 1 3 + p 2 5 = n has a solution with primes p 1, p 2 and a P 15-number x, and that for every sufficiently large even n, the equation x +p 1 2 + p 2 3 + p 3 4 = n has a solution with primes p i and a P 2-number x.

On the sum of four cubes

On Warings problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E ( N ) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E ( N )≪1.

Relations between exceptional sets for additive problems

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring�s problem for cubes, we show, in particular, that the number of positive integers not exceeding N that fail to have a representation as the sum of six cubes of natural numbers is O(N3/7).