Note on the sum of cubes of primes and an almost prime

Abstract.

Let P _s denote the natural numbers that are the product of at most s prime numbers, and let p, q, r denote prime numbers. In connection with the Waring-Goldbach problem for cubes, J. Brüdern proved that almost all numbers n are written in the form n = P ₄ ³ + p ³ + q ³ + r ³ (Ann. Scient. Éc. Norm. Sup., 1995). In this note, it is shown by combining the argument of Brüdern with the reversal rôle technique in the sieve theory that one can replace the subscript 4 by 3. More precisely, all n≤N with some local conditions, except for O (N ( log 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 at once that every sufficiently large even number can be written as a sum of cubes of seven primes and a P ₃.