ζk is one of the most important constant in the siève methods.Tlus paper gives the relatively accurate lower bound and upper bound on it,that is,3<sup>-1/k</sup>ck,where c=1.22... and k】16.
Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(...Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].展开更多
The nine typical Shanghai soils are usually silty clay or clay,which appears inconsistent with their low clay content in the relevant publications.The literature review shows that the documented clay content of Shangh...The nine typical Shanghai soils are usually silty clay or clay,which appears inconsistent with their low clay content in the relevant publications.The literature review shows that the documented clay content of Shanghai soils ranges from 0%to 30.8%by weight.This inconsistency may stem from two factors:(1)the Shanghai soil classification system relies solely on the plasticity index for soil naming;and(2)the conventional steel sieving method cannot separate the clay from the fine soils(clay and silt mixtures).This paper aims to accurately determine the clay content in Shanghai soils.It uses nylon cloth sieves with apertures ranging from 0.063 mm to 0.0008 mm and completely separates the clay particles from the fine soils.The nine typical Shanghai soils are tested and sieved into distinct subgroups of clay,silt,sand,and gravel particles.Results demonstrate clay content ranges from 18.99%to 79.33%,substantially higher than literature values and consistent with their names of either silty clay or clay.Macro,micro,and scanning electron microscope(SEM)images reveal effective separation of clay,silt,sand,and gravel particles.The clay exhibits cohesive properties,while the silt,sand,and gravel comprise clean,non-cohesive individual particles.The clay and silt fractions are confirmed to be within their respective sieving limits by SEM-based particle size measurements.Additionally,Atterberg limits testing highlights the high plasticity of the clay particles and the non-plastic nature of the silt particles.展开更多
Using the circle method and sieves, the author proves that a positive proportion of positive integers can be represented as the sum of four cubes of primes.
Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+...Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.展开更多
It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 ＋ p2, p1 ∈ Pγ is solvable, with 13/15 〈γ≤ 1, where Pγ = {p ｜p = [m^1/γ], for integer m and prime p} is t...It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 ＋ p2, p1 ∈ Pγ is solvable, with 13/15 〈γ≤ 1, where Pγ = {p ｜p = [m^1/γ], for integer m and prime p} is the set of the Piatetski-Shapiro primes.展开更多
Under the Generalized Riemann Hypothesis, it is proved that for any integer \%k≥770\% there is \%N\-k>0\% depending on \%k\% only such that every even integer ≥\%N\-k\% is a sum of two odd prime numbers and \%k\...Under the Generalized Riemann Hypothesis, it is proved that for any integer \%k≥770\% there is \%N\-k>0\% depending on \%k\% only such that every even integer ≥\%N\-k\% is a sum of two odd prime numbers and \%k\% powers of 2.展开更多
Let N be a sufficiently large even integer.Let p denote a prime and P2 denote an almost prime with at most two prime factors.In this paper,it is proved that the equation N = p + P2(p ≤ N0.945) is solvable.
文摘ζk is one of the most important constant in the siève methods.Tlus paper gives the relatively accurate lower bound and upper bound on it,that is,3<sup>-1/k</sup>ck,where c=1.22... and k】16.
基金Supported by NSFC (Nos.12471009,12301006,12001047,11901566)Beijing Natural Science Foundation (No.1242003)National Training Program of Innovation and Entrepreneurship for Undergraduates(No.202307011)。
文摘Let Pr denote an almost-prime with at most r prime factors,counted according to multiplicity.In this paper,it is proved that,for every sufficiently large even integer N,the equation N=x^(2)+p_(2)^(2)+p_(3)^(3)+p_(4)^(3)+p_(5)^(5)+_6^(5)is solvable with being an almost-prime P_(6) and the other variables primes.This result constitutes an enhancement upon the previous result of Hooley[Recent Progress in Analytic Number Theory,Vol.1(Durham,1979),London:Academic Press,1981,127-191].
基金supported by the Research Grant Council of the Hong Kong Special Administrative Region,China(Grant Nos.HKU 17207518 and R5037-18).
文摘The nine typical Shanghai soils are usually silty clay or clay,which appears inconsistent with their low clay content in the relevant publications.The literature review shows that the documented clay content of Shanghai soils ranges from 0%to 30.8%by weight.This inconsistency may stem from two factors:(1)the Shanghai soil classification system relies solely on the plasticity index for soil naming;and(2)the conventional steel sieving method cannot separate the clay from the fine soils(clay and silt mixtures).This paper aims to accurately determine the clay content in Shanghai soils.It uses nylon cloth sieves with apertures ranging from 0.063 mm to 0.0008 mm and completely separates the clay particles from the fine soils.The nine typical Shanghai soils are tested and sieved into distinct subgroups of clay,silt,sand,and gravel particles.Results demonstrate clay content ranges from 18.99%to 79.33%,substantially higher than literature values and consistent with their names of either silty clay or clay.Macro,micro,and scanning electron microscope(SEM)images reveal effective separation of clay,silt,sand,and gravel particles.The clay exhibits cohesive properties,while the silt,sand,and gravel comprise clean,non-cohesive individual particles.The clay and silt fractions are confirmed to be within their respective sieving limits by SEM-based particle size measurements.Additionally,Atterberg limits testing highlights the high plasticity of the clay particles and the non-plastic nature of the silt particles.
基金Project supported by the National Natural Science Foundation of China (No.10041004) and the ThansCentury naming Programme Foun
文摘Using the circle method and sieves, the author proves that a positive proportion of positive integers can be represented as the sum of four cubes of primes.
基金supported by the National Natural Science Foundation of China(Nos.10771103,10801075)the Natural Science Foundation of Huaihai Institute of Technology(No.KQ10002)
文摘Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.
基金Project supported by the Foundation of Shandong Provincial Education Department in China (No.03F06)the Grant for Doctoral Fellows in Shandong Finance Institute
文摘It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 ＋ p2, p1 ∈ Pγ is solvable, with 13/15 〈γ≤ 1, where Pγ = {p ｜p = [m^1/γ], for integer m and prime p} is the set of the Piatetski-Shapiro primes.
基金ProjectpartiallysupportedbyRGCResearchGrant (No .HKU 712 2 / 97P)andPost DoctoralFellowshipoftheUniversityofHongKong .
文摘Under the Generalized Riemann Hypothesis, it is proved that for any integer \%k≥770\% there is \%N\-k>0\% depending on \%k\% only such that every even integer ≥\%N\-k\% is a sum of two odd prime numbers and \%k\% powers of 2.
基金Project supported by the National Natural Science Foundation of China(No.11071186)the Science Foundation for the Excellent Youth Scholars of Shanghai(No.ssc08017)the Doctoral Research Fund of Shanghai Ocean University
文摘Let N be a sufficiently large even integer.Let p denote a prime and P2 denote an almost prime with at most two prime factors.In this paper,it is proved that the equation N = p + P2(p ≤ N0.945) is solvable.
