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postgraduate thesis: On linear equations in primes and powers of two
Title  On linear equations in primes and powers of two 

Authors  
Advisors  Advisor(s):Tsang, KM 
Issue Date  2013 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Kong, Y. [孔亚方]. (2013). On linear equations in primes and powers of two. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5053376 
Abstract  It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the GoldbachLinnik 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. HeathBrown 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)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the rowspace of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation
{█({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^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and B2 < BB1, where B = max1≤j≤4(2, aj), can be represented as
{█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(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 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤
Moreover 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)〗)/log0.97580584.0285], (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)〗^(1) ×〖(3B)〗^(1) (10.000001) 〖(3B)〗^(14), with B = max1≤j≤4(2, a2j), and
F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 ))
with
f(a_2i) = {█(4414.15h (a_211)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤
for i = 1, 2, and h(n) =∏_(pn,p>2)▒(p1)/(p2). This result, if without the powers of
two, can make up some of the cases excluded in Green and Tao’s paper. 
Degree  Doctor of Philosophy 
Subject  Numbers, Prime Algebras, Linear. Number theory. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/188257 
HKU Library Item ID  b5053376 
DC Field  Value  Language 

dc.contributor.advisor  Tsang, KM   
dc.contributor.author  Kong, Yafang.   
dc.contributor.author  孔亚方.   
dc.date.accessioned  20130827T08:02:51Z   
dc.date.available  20130827T08:02:51Z   
dc.date.issued  2013   
dc.identifier.citation  Kong, Y. [孔亚方]. (2013). On linear equations in primes and powers of two. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5053376   
dc.identifier.uri  http://hdl.handle.net/10722/188257   
dc.description.abstract  It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the GoldbachLinnik 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. HeathBrown 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)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the rowspace of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation {█({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^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and B2 < BB1, where B = max1≤j≤4(2, aj), can be represented as {█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(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 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤ Moreover 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)〗)/log0.97580584.0285], (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)〗^(1) ×〖(3B)〗^(1) (10.000001) 〖(3B)〗^(14), with B = max1≤j≤4(2, a2j), and F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 )) with f(a_2i) = {█(4414.15h (a_211)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤ for i = 1, 2, and h(n) =∏_(pn,p>2)▒(p1)/(p2). This result, if without the powers of two, can make up some of the cases excluded in Green and Tao’s paper.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  Creative Commons: Attribution 3.0 Hong Kong License   
dc.source.uri  http://hub.hku.hk/bib/B50533769   
dc.subject.lcsh  Numbers, Prime   
dc.subject.lcsh  Algebras, Linear.   
dc.subject.lcsh  Number theory.   
dc.title  On linear equations in primes and powers of two   
dc.type  PG_Thesis   
dc.identifier.hkul  b5053376   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5053376   
dc.date.hkucongregation  2013   