Representation of even integers by cubes of primes and powers of 2 (Q5948325)

The authors investigate the representation of even integers by cubes of primes and powers of \(2\). Three theorems and some lemmas are proven. Their main result is: Theorem. There exist positive absolute constants \(k\) and \(N_0\) such that each even integer \(N\geq N_0\) can be written as \[ N=p_1^3+p_2^3+\ldots +p_8^3+2^{\upsilon_1}+2^{\upsilon_2}+\ldots +2^{\upsilon_k}, \] where \(p\) and \(\upsilon\), with subscripts, denote a prime number and a positive integer, respectively. The proof depends essentially on the circle method and sieve methods.