NOTE ON THE SUM OF CUBES OF PRIMES AND AN ALMOST PRIME

Let P-s denote the natural numbers that are the product of at most s p rime numbers, and let p, q, r denote prime numbers. In connection with the Waring-Goldbach problem for cubes, J. Brudern proved that almost all numbers it are written in the form n = P-4(3) + p(3) + q(3) +r(3) (Am. Scient. Ec. Norm. Sup., 1995). In this note, it is shown by combi ning the argument of Brudern with the reversal role technique in the s ieve theory that one can replace the subscript 4 by 3. More precisely, all n less than or equal to N with some local conditions, except for O(N(logn N)(-A)) exceptions, can be written in the form n = P-3(3) + p (3) + q(3) + r(3), where A is any fixed positive number. This yields a t once that every sufficiently large even number can be written as a s um of cubes of seven primes and a P-3.