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
In this paper,the authors investigate exceptional sets in the Waring-Goldbach problem for unlike powers.For example,estimates are obtained for sufficiently large integers below a parameter subject to the necessary loc...In this paper,the authors investigate exceptional sets in the Waring-Goldbach problem for unlike powers.For example,estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not have a representation as the sum of a square of prime,a cube of prime and a sixth power of prime and a k-th power of prime.These results improve the recent result due to Brüdern in the order of magnitude.Furthermore,the method can be also applied to the similar estimates for the exceptional sets for Waring-Goldbach problem for unlike powers.展开更多
Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_...Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_(1)^(3)+p_(2)^(4)+p_(3)^(4)+p_(5)^(4)+p_(6)^(4)+p_(7)^(4)+p_(8)^(4)+p_(9)^(4)+p_(10)^(4),where p1,p2,…,P_(10)are prime numbers.展开更多
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that wi...In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.展开更多
In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers ...In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.展开更多
We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth...We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes,which improves the previous results due to A.V.Kumchev[Canad.J.Math.,2005,57:298–327]and Z.X.Liu[Int.J.Number Theory,2012,8:1247–1256].展开更多
In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due...In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.展开更多
Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation n=x^3＋p1^3＋p2^3＋p3^3＋p...Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation n=x^3＋p1^3＋p2^3＋p3^3＋p4^3＋p1^3＋p5^3＋p6^3＋p7^3 has solutions in primes pi with x being a P6. This result constitutes a refinement upon that of Hooley C.展开更多
It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous re...It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.展开更多
In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent...In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.展开更多
It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> s...It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’s theorem on the quadratic Waring-Goldbach problem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.
We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 ＋p2^3 ＋... ＋p9^3 with [pj - （N/9）^1/31 ≤ N^（1/3）-θ, where pj, j = 1,2,...,9, are primes and θ = （1/51） -ε.
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 National Natural Science Foundation of China(No.11771252)the Natural Science Foundation of Fujian Province(Nos.2024J01805,2022J02046).
文摘In this paper,the authors investigate exceptional sets in the Waring-Goldbach problem for unlike powers.For example,estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not have a representation as the sum of a square of prime,a cube of prime and a sixth power of prime and a k-th power of prime.These results improve the recent result due to Brüdern in the order of magnitude.Furthermore,the method can be also applied to the similar estimates for the exceptional sets for Waring-Goldbach problem for unlike powers.
文摘Let N be a sufficiently large integer.In this paper,it is proved that with at most O(N17/18+ε)exceptions,all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_(1)^(3)+p_(2)^(4)+p_(3)^(4)+p_(5)^(4)+p_(6)^(4)+p_(7)^(4)+p_(8)^(4)+p_(9)^(4)+p_(10)^(4),where p1,p2,…,P_(10)are prime numbers.
基金The author is supported by Post-Doctoral Fellowsbip of The University of Hong Kong.
文摘In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2+log κ/log2, x≥2 and α=a/q+ λ subject to (a, q) = 1, 1≤a≤q, and λ ∈ R. Then As an application, we prove that with at most O(N2/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.
基金supported by the National Natural Science Foundation of China(No.11771333)
文摘In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.
基金The first author was supported by the Scientific Research Project of the Education Department of Fujian Province(Grant No.JAT190370)the Natural Science Foundation of Fujian Province(Grant No.2020J05162)+1 种基金The second author was supported by the National Natural Science Foundation of China(Grant No.11871367)the Natural Science Foundation of Tianjin City(Grant No.19JCQNJC14200).
文摘We consider exceptional sets in the Waring-Goldbach problem for fifth powers.For example,we prove that all but O(N^(131/132))integers satisfying the necessary local conditions can be represented as the sum of 11 fifth powers of primes,which improves the previous results due to A.V.Kumchev[Canad.J.Math.,2005,57:298–327]and Z.X.Liu[Int.J.Number Theory,2012,8:1247–1256].
基金Supported by Post-Doctoral Fellowship of The University of Hong KongThe National Natural Science Foundation(Grant No.10571107)Supported by a grant from the Research Grant Council of Hong Kong(Project No.HKU7028/03P)
文摘In this paper, it is proved that with at most O（N65/66） exceptions, all even positive integers up to N are expressible in the form p^2 2＋p^3 3＋p^4 4＋p^5 5. This improves a recent result O（N19193/19200＋ε） due to C. Bauer.
基金supported by the National Natural Science Foundation of China(No.11201107)the China Scholarship Council
文摘Let Pr denote an almost prime with at most r prime factors, counted according to multiplicity. In the present paper, it is proved that for any sufficiently large even integer n, the equation n=x^3＋p1^3＋p2^3＋p3^3＋p4^3＋p1^3＋p5^3＋p6^3＋p7^3 has solutions in primes pi with x being a P6. This result constitutes a refinement upon that of Hooley C.
基金Supported by The National Science Foundation(Grants #10125101 and #10131010)by a Ministry of Education Major Grant Program in Sciences and Technology
文摘It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.
基金supported by the National Natural Science Foundation of China(Grant Nos.10125101&10531060)a Major Grant Program in Science and Technology by the Ministry of EducationTianyuan Mathematics Foundation(Grant No.10526028).
文摘In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.
基金Supported by MCSEC and the National Natural Science Foundation (Grant No. 19701019) Supported by MCSFC and the National Natural Science Foundation
文摘It is proved that every large integer N≡5（mod24）can be written as N=p<sub>1</sub><sup>2</sup>+…+p<sub>5</sub><sup>2</sup> with each prime p<sub>j</sub> satisfying |p<sub>J</sub>-（N/5|）<sup>1/2</sup>≤N<sup>11/23</sup>.This gives a short interval version of Hua’s theorem on the quadratic Waring-Goldbach problem
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.
文摘We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 ＋p2^3 ＋... ＋p9^3 with [pj - （N/9）^1/31 ≤ N^（1/3）-θ, where pj, j = 1,2,...,9, are primes and θ = （1/51） -ε.
