On the GoldbachLinnik Problem and Its Extensions 

Author  LiuZhiXin 
Tutor  LiuJianYa 
School  Shandong University 
Course  Basic mathematics 
Keywords  Circle method WaringGoldbach problem A power of 2 
CLC  O156.1 
Type  PhD thesis 
Year  2011 
Downloads  23 
Quotes  0 
Famous Goldbach conjecture can be expressed as: (A) each greater than or equal to 9 odd into three odd prime numbers and can be expressed; (B) each even number greater than or equal to 6 can be expressed as the sum of two odd the primes and apparently, conjecture (A) conjecture (B) direct inference .1937, Vinogradov [70] basically solved the conjecture (A), he proves that every sufficiently large odd number can be expressed as three odd primes sum, this result is also known as the three prime number theorem in this article, Goldbach conjecture refers specifically to conjecture (B). ways as a Goldbach conjecture proved in 1951, Linnik [30] under the generalized Riemann hypothesis, the two years after [31] unconditionally proved each sufficiently large even can be expressed as a power of 2 and of two prime numbers, and can control the number of, i.e., N = p1 p2 2V1 ... 2vK1, this problem is called \GoldbachLinnik problem \In fact, the number of such integers in the range [1, N] O (logk N) in GoldbachLinnik problem of significance is that, although we can not prove the Goldbach conjecture, but can prove Goldbach conjecture and then paste a sparse chapter finer calculation of the main range and I circle method interval to improve the above results. Theorem 1.1. every sufficiently large even number can be represented as two prime numbers with no more than 12 and a power of 2 . ieK1 ≤ 12. in the first chapter, we also discussed the Romanoff problems, generalized twin primes and other related issues. linked to Hua [19] the five primes square and Theorem Lagrangc foursquare and theorems, and Inspired by the Linnik the GoldbachLinnik problem, Liu, Liao Mingzhe and Zhan Tao [38] proved that every sufficiently large even number can be expressed as a the four primes square with a power of 2 and, N = p12 p22 p32 p42 2v1 ... 2vk2. because this issue was first proposed by Gallagher, so is called \as the GoldbachLinnik and the LinnikGallaghcr mixed problem] fixed and improved., Liu, Liao Mingzhe and Zhan Tao [38] proved that every sufficiently large odd number can be expressed as a prime number, the two primes square with 2 the nine primes cubic Theorem higher powers of the form of a conjecture under conditions to further improve the value of the K3. Similarly, we can also consider the problem of GoldbachLinnik LinnikGallagher. linked to Hua [19] Liu Asian and Liao Mingzhe [34] proved that every sufficiently large even number can be expressed as the eight primes cubic power of 2, and N = p13, p23 ... p83 2v1 2v2 ... 2vk4 In the second chapter , the first time we are given a K4 allowable value. Theorem 2.1. every sufficiently large even number can be expressed as the eight primes cubic more than 358 and a power of. ieK4 ≤ 358 recently, Lvguang Shi and the author [49] consider primes of unequal time power 2 square power of and, we prove that each sufficiently large even number can be expressed as a prime number, the square of a prime number, two primes cubic with two of the party power each sufficiently large sum of N = p1 p22 of p33 to p43 2v1 2v2 ... 2vks further, we set the value K5 allowable, K5 ≤ 161. in the third chapter, we further improved the above results. Theorem 3.2. even number can be expressed as a prime number, the square of a prime number, the two prime cubes with no more than 124 and a power of the ieK5 ≤ 124. Similarly, as the GoldbachLinnik, LinnikGallagher problems and mixed problem of the eight primes cubic power of 2 of the problem, Lvguang Shi and the author [48] proved that every sufficiently large odd number can be expressed as a prime number, and four primes cubic power of 2, N = p1 p23 p33 P43 p53 2v1 2v2 ... 2ck6. further, we set K6 allowable value and for K6 ≤ 106. addition, each sufficiently large even number can be expressed as the square of two prime numbers, four primes cubic, with a power of 2 and N = p12 p22 of p33 p43 of p53 p63 2v1 2v2 ... 2vk7 further, we set K7 allowable value for K7 ≤ 211. in the fourth chapter, we further improved the above result. Theorem 4.2. every sufficiently large odd number can be expressed as a prime number, the four primes cubic and not more than 97 power of 2, and, ieK6 ≤ 97. Theorem 4.5 each sufficiently large even into two prime numbers, square, the four primes cubic 136 and a power of not more than In ieK7 ≤ 136. in the fifth chapter, we consider the square of a prime number, prime cubes 2 square power and the other two results. N = p12 p22 p32 p43 of p53 2v1 ... 2vk8, N = p12 p23 p73 2v1 2vk9. Rather, we prove Theorem 5.1 every sufficiently large odd number can be expressed as three primes squared, two primes cubic with more than 172 and a power of, ieK8 Theorem ≤ 172. 5.2. every sufficiently large odd number can be expressed as the square of a prime number, the six primes cubic with more than 111 and a power of 2, ieK9 ≤ 111.