A Waring-Goldbach type problem for mixed powers
Let \(R_4(N)\) denote the number of ways of writing \(N\) in the form \[ N=x^2+p_2^2+p_3^3+p_4^4+p_5^4+p_6^4, \] where \(p_j\)'s are primes and \(x\) is a \(P_6\) (a \(P_r\) means an integer having at most \(r\) prime factors, counted according to multiplicity). The paper is mostly devoted to proving that \(R_4(N)\gg N^{12/13}(\log N)^{-6}\) for all sufficiently large integers \(N\) not congruent to 2 modulo 3. The latter lower bound agrees with the expected order of magnitude of \(R_4(N)\), and the congruence condition \(N\not\equiv\pmod{3}\) arises naturally from the fact that \(p^2\equiv1\) and \(p^3\not\equiv0\pmod{3}\) for all primes \(p\neq3\). The proof applies the Hardy-Littlewood method and a linear sieve with switching principle. The author also asserts that the methods of the paper may lead to the following theorem: Define \(R_5(N)\) similarly, replacing \(p_6^4\) and \(P_6\) in the definition of \(R_4(N)\) by \(p_6^5\) and \(P_9\), respectively. Then for all sufficiently large even \(N\), one has \(R_5(N)\gg N^{31/30}(\log N)^{-6}\).
scientific article; zbMATH DE number 6343977
scientific article