Goldbach–Linnik type problems with one prime square and six prime cubes

Let p1, p2, …, p7 be prime numbers. In this paper, we first show that when k1 ≥ 49, any sufficiently large odd integer can be represented as the sum of a prime square, six prime cubes and k1 powers of 2. Furthermore, we prove that for k2 ≥ 73, every pair of sufficiently large odd integers can be expressed as a pair of equations involving a prime square, six prime cubes and k2 powers of 2.