| The famous Goldbach Conjecture can be stated as:(A) Every odd number which is equal to or greater than 9 is the sum of three odd primes;(B) Every even number which is equal to or greater than 6 is the sum of two odd primes.Obviously, Conjecture (B) implies Conjecture (A). In 1937, Conjecture (A) was almost solved by Vinogradov [70], who proved that every sufficiently large odd number can be written as the sum of three odd primes. The Conjecture (A) is also known as the three prime theorem. In this dissertation, Goldbach Conjecture exactly means Conjecture (B).As an approach to prove Goldbach Conjecture, Linnik [30] proved under GRH in 1951, and two years later unconditionally [31] that every sufficiently large even integer can be written as a sum of two primes and a bounded number of powers of 2, N＝p1＋p2＋2v1＋…＋2vk1. This problem is called "Goldbach-Linnik problem" (sec [55]) or "almost Goldbach prob-lem" (sec [41]). We call "Goldbach-Linnik problem" to "almost Goldbach problem", since the set {2v1+...+2vk:Vj≥0} is very thin. In fact, there are only O(logk N) such integers in the interval [1, N]. The significance of the Goldbach-Linnik problem is that, although currently we can not prove Goldbach Conjecture, but it is true if we paste a thin set in Goldbach Conjecture. Obviously, The best estimate K1=0 and the Goldbach Conjecture arc equivalent.Many authors determined and improved the values of K1 under GRH or uncon-ditionally (see [35], [36], [37], [26], [72], [27], [18], [56] etc.). The best result about the value of K1 is due to Heath-Brown and Puchta [18], who proved K1≤13.In Chapter 1 of this dissertation, we give an improvement for the previous results by much more careful estimate on both major arcs and minor arcs.Theorem 1.1. Every sufficiently large even integer can be written as the sum of two primes and 12 powers of 2. i.e. K1≤12.Romanoff problem, Generalized twin-prime problem and other related problems arc also discussed in Chapter 1.In view of Hua’s theorem on five prime squares [19] and Lagrange’s theorem of four squares, as an extension of Goldbach-Linnik problem, Liu, Liu and Zhan [38] proved that every sufficiently large even integer can be written as the sum of four squares of primes and powers of 2, N=P12＋p22＋p23＋p24＋2v1＋…＋2vk2. This problem was first asked by Gallagher, so is known as "Linnik-Gallaghcr problem" (see [41]). The acceptable values for K2 have been obtained in [32], [39] and [28].As the hybrid problem of Goldbach-Linnik problem and Linnik-Gallaghcr problem, Liu, Liu and Zhan [38] proved every sufficiently large odd integer can be written as the sum of a prime, two squares of primes and powers of 2, i.e. N= P1+p22+p23+2v1+...+2vk3, where the acceptable values for K3 have been obtained in [43], [29] and [51].In Chapter 1, we further give an improvement for the value of K3 under a conjec-ture of Generalized twin-prime problem.In view of Hua’s theorem on nine prime cubes [19], we can consider the higher powers of primes for Goldbach-Linnik problem and Linnik-Gallaghcr problem. Liu and Liu [34] proved every sufficiently large even integer can be written as the sum of eight cubes of primes and powers of 2, N=p31+p32+...+p38+2v1+2v2+...+2vk4. In Chapter 2,we firstly give the acceptable value of K4.Theorem 2.1. Every sufficiently large even integer can be written as the sum of eight cubes of primes and 358 powers of 2,i.e.K4≤358.Recently,Lu and the author[49]considered representation of integers by unlike powers of primes and powers of 2.We proved that every suffciently large even intcgcr can be written as the sum of a prime,a square of prime,two cubes of primes and poeers of 2, N=p1+p22+p33+p43+2v1+2v2+…+2vk5. Furthermore,we gave the acceptable value of K5,that is K5≤161.In Chapter 3,we give an improvement for the previous result.Theorem 3.2. Every suffciently large even integer can be written as the sum of a prime,a square of prime, two cubes of primes and 124 powers of 2,i.e.K5≤124.Similarly,as the hybrid problems of Goldbach-Linnik problem,Linnik-Gallagher problem and representation of integers by eight cubes of primes and powers of 2,Lu and the author[48]proved that every suffciently large odd integer can be written as the sum of a prime,four cubes of primes and powers of 2, N=p1+p23+p33+p43+p53+2v1+2v2+…+2vx6. Furthermore,we gave the acceptable value of K6,that is K6≤106.In addition,every suffciently large even integer can be written as the sum of two squares of prime,four cubes of primes and powers of 2, N=p12+p22+p33+p43+p53+p63+2v1+2v2+…+2vk7. Furthermore,we gave the acceptable value of K7,that is K7≤211.In Chapter 4,we give an improvement for the previous results.Theorem 4.2. Every sufficiently large odd integer can be weitten as the sum of a prime,four cubes of primes and 97 powers Of 2,i.e.K6≤97.Theorem 4.5. Every suffciently large even integer can be written as the sum of two squares of prime,four cubes of primes and 136 powers of 2,i,e.K7≤136.In Chapter 5,we consider two results about squares and cubes of primes and powers of 2. N=p12+p22+p32+p43+p53+2v1+…+2vk8, N=p12+p23+...+p73+2v1+.+2vk9 Exactly,we proveTheorem 5.1 Every sufficiently large odd integer can be written as the sum of three squares of prime, two cubes of primes and 172 powers of 2,i.e.K8≤172.Theorem 5.2 Every suffciently large odd integer can be written as the sum of a squares of prime, sir cubes of primes and 111 powers of 2,i.e.K9≤111. |
