Two Generalizations Of Hua’s Theorem

The distribution of prime numbers and Diophantine equation are two basic problems in the study of number theory.The prime solutions of Diophantine equation are the intersection of these two basic problems.Hua made an important contribution to the study of the prime solutions of Diophantine equation and established many profound theorems.In this paper,we mainly consider two generalizations of Hua’s theorems,the main results are as follows.Firstly,we prove that the values taken by real linear combinations of two primes and one prime square,i.e.,λ1p1+λ2p2+λ3p32 can be arbitrarily close to any real number,where λ1,λ2,λ3 are non-zero real numbers,not all of the same sign and λ2/λ3 is irrational.Secondly,for every sufficiently large odd integer N satisfying N(?)1(mod 3),we prove that N can be represented as the sum of a small prime and the squares of four primes,i.e.,N=p+p12+p22+p32+p42 with p≤N49/144.The structure of this paper is organized as follows.In Chapter 1,we describe the background and main results of this paper.In Chapters 2 and 3,we give the outlines of proofs of Theorems 1 and 2,respectively.In Chapter 3,we give some preliminary lemmas which are needed to prove Theorems 1 and 2.In Chapters 5 and 6,we complete the proofs of Theorems 1 and 2,respectively.