On Linear Equations in Primes and Powers of Two - Bookswagon
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On Linear Equations in Primes and Powers of Two

On Linear Equations in Primes and Powers of Two


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This dissertation, "On Linear Equations in Primes and Powers of Two" by Yafang, Kong, 孔亚方, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the Goldbach-Linnik problem, that is, representation of an even integer in the form of two odd primes and powers of two, as its approximate problem. The theme of my research is on linear equations in primes and powers of two. Precisely, there are two cases: one pair of linear equations in primes and powers of two, and one class of pairs of linear equations in primes and powers of two, in this thesis. In 2002, D.R. Heath-Brown and P.C. Puchta obtained that every sufficiently large even integer is the sum of two odd primes and k powers of two. Here k = 13, or = 7 under the generalized Riemann hypothesis. In 2010, B. Green and T. Tao obtained that every pair of linear equations in four prime variables with coefficients matrix A = (a_ij)st with s {█({B1 = p1 ] p2 + 2v1 + 2v2 + - - - + 2vk, @B2 = p3 + p4 + 2v1 + 2v2 + - - - + 2vk, )┤ (1) is solvable, where p1, - - -, p4 are odd primes, each vi is a positive integer, and the positive integer k >= 63 or >= 31 under the generalized Riemann hypothesis. Note that, in 1989, M.C. Liu and K.M. Tsang have obtained that subject to some natural conditions on the coefficients, every pair of linear equations in five prime variables is solvable. Therefore one class of pairs of linear equations in four prime variables with special coefficient matrix and powers of two is considered. Indeed, it is deduced that every pair of integers B1 and B2 satisfying B1 ≡ 0 (mod 2), 3BB1 > e DEGREES(eB DEGREES48 ), B2 ≡ ∑_1 DEGREES4▒= 1 DEGREES(a_i ) (mod 2) and B2 {█(B1 = 〖p1〗_1 + p2 + 2 DEGREES(v_1 ) + 2 DEGREES(v_2 )+ - - - + 2 DEGREES(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2 DEGREES(v_1 )+ 2 DEGREES(v_2 )+ - - - + 2 DEGREES(v_k ) )┤ (2) with k being a positive integer. Here p1, - - - p4 are odd primes, each 〖v 〗_iis a positive integer and the integral coefficients ai (i = 1, 2, 3, 4) satisfy {█((〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) = 1, @〖a 〗_1 〖a 〗_2Moreover it is calculated that the positive integer k >= g(〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) where g(〖a 〗_21- 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = [(log⁡〖G(〖a 〗_21, ..., 〖a 〗_24 〗)-log⁡〖F (〖a 〗_21, ..., 〖a 〗_24)〗)/log0.975805-84.0285], (3) G(〖a 〗_21, 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = (min(1/(a_24 ), 1/(a_23 )) - (〖a〗_(21 )- a_22 )/(〖a_23 a〗_24 ) 〖(3B)〗 DEGREES(-1) 〖(3B)〗 DEGREES(-1) (1-0.000001)- 〖(3B)〗 DEGREES(-1-4), with B = max1F(a_21, ..., a_24) = √(f(a_21)f〖(a〗_22 )) with f(a_2i) = {█(4414.15h (a_21-1)+5.088331 if a_211@59.8411 if a_21=1, )┤ for i = 1, 2, and h(n) =∏_(pn, p>2)▒(p-1)/(p-2). This result, if without the powers of two, can make up some of the cases excluded in


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Product Details
  • ISBN-13: 9781361318300
  • Publisher: Open Dissertation Press
  • Publisher Imprint: Open Dissertation Press
  • Height: 279 mm
  • No of Pages: 118
  • Weight: 567 gr
  • ISBN-10: 1361318309
  • Publisher Date: 26 Jan 2017
  • Binding: Hardback
  • Language: English
  • Spine Width: 8 mm
  • Width: 216 mm


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