Exceptional sets for sums of five and six almost equal prime cubes

Abstract

We investigate the exceptional sets of natural number n which can be represented as sums of five and six cubes of almost equal primes, i.e. $${n=p_1^3+\cdots+p_s^3}$$ (s=5,6). It is established that almost all natural numbers n subject to certain congruence conditions have the above representation with $${|p_j-(n/s)^{\frac{1}{3}}|\leq n^{\theta_s/3+\varepsilon}}$$ ( $${1\leq j\leq s}$$ ), where $${\theta_5=8/9+\varepsilon}$$ and $${\theta_6=5/6+\varepsilon}$$ .