On representation of integers by sums of a cube and three cubes of primes (Q2491064)

Let \(N\) be a large positive integer and let \(E(N)\) denote the number of positive integers \(n\leq N\) that cannot be written in the form \[ n=m^3+p_2^3+p_3^3+p_4^3 \] with positive integer \(m\) and primes \(p_j\). For any positive number \(\varepsilon\) the authors prove that \(E(N)\ll N^{17/18+\varepsilon}\).