Two results on powers of 2 in Waring-Goldbach problem
Following the line of investigation of \textit{Yu. V. Linnik} [Tr. Mat. Inst. Steklova 38, 152--169 (1951; Zbl 0049.31402)] that every large even integer is the sum of two odd primes and a fixed number of powers of two, many authors have improved Linnik's result and also generalized it to similar representations with higher powers of primes. In the present paper the authors prove the following two results: (1) every large odd integer is the sum of one prime, four cubes of primes and 106 powers of two, (2) every large even integer is the sum of two squares of primes, four cubes of primes and 211 powers of two. Furthermore, under the conjecture of an asymptotic formula predicted by Hardy-Littlewood on the number of representations of non-zero number as the difference of two primes, the 106 in (1) can be improved to 74. The method of proof uses the circle method. On the minor arcs, exponential sums over squares and cubes of primes are estimated by recent results of \textit{A. V. Kumchev} [Mich. Math. J. 54, No. 2, 243--268 (2006; Zbl 1137.11054)].
scientific article; zbMATH DE number 5859976
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