A Generalization Of Three Primes Theorem

In 1937, I. M. Vinogradov proved the famous Three primes theorem, which states that for every sufficiently large odd integer N, whereThree primes theorem is one of the most important conclusions in analytic number theory, which people have been going deep into studying. For example, Pan, Zhan and Jia C. H. studied Three primes theorem in short intervals.In 1953, Piatetski-Shapiro first studied primes in the squence [nc](c>0, n= 1,2,…).Let Piatetski-Shapiro proved for 155/28, then for some constant c0 which depends on c,d. Sirota proved that (2) holds for for an absolute constant c1.E. Wirsing first studied Three primes theorem in thin subsets of primes. He showed there exists a subset of primes S with the property that such that every sufficiently large odd integer N is a sum of three elements of S. But people know little about S.In 1992, A. Balog and J. Friedlander first studied the Waring-Goldbach problem for odd numbers in Piatetski-Shapiro primes set Pγ={p|p=[n1/γ],n∈N}. They proved that the equation is soluable for 20/21<γ<1. Later, J. Rivat extended the rang 20/21<γ<1 to 188/199<γ<1. C. H. Jia applied the sieve method to extend the rang 20/21<γ<1 to 15/16<γ<1.In this paper, we shall investigate the prime equation where N is a sufficiently large odd integer with 1117/59, A>0 is an arbitrary positive constant, then we have...