Waring-Goldbach problem involving cubes of primes (Q2663070)

Using the circle method the authors prove two main results: (a) Every sufficiently large integer is a sum of one prime number, four cubes of primes, and fifteen powers of \(2\). Assuming the Riemann hypothesis the number of powers of two is reduced to seven. (b) At least \(33/400\) of all positive integers congruent to \(r \pmod{q}\) are sums of four cubes of prime numbers. Here, \(q/3\) equals the product of all primes less than \(1000\), \(r\) is a positive even integer such that \(r \in \{2,4,5,7\} \pmod{9}\), and \(r \in \{2,3,4,5\} \pmod{7}\).