A SIEVE APPROACH TO THE WARING-GOLDBACH PROBLEM .1. SUMS OF 4 CUBES

The problem of representing integers as the sum of k-th powers of prim es is known as the Waring-Goldbach problem. Traditionally results on t his problem are obtained by reference to auxiliary estimates from the ''ordinary'' Waring problem, which are then combined with Vinogradov's estimates for exponential sums over primes. Here we describe an alter native approach, based on the linear sieve and the circle method, and show that almost all natural numbers n = 4 mod 24 can be written as n = p(1)(3) + p(2)(3) + p(3)(3) + x(3) where p(1), p(2), p(3) are primes , and x has at most four prime factors. Our method has the advantage t hat one can deal with fewer variables than is possible by Vinogradov's method, but sometimes detects an ''almost prime'' rather than a prime .