华林-哥德巴赫问题:两个平方,两个立方和两个四次方

令P_r表示素因子不超过r的殆素数,按重数计.作者证明了对于充分大的偶数N,方程N=x~2+p_1~2+p_2~3+p_3~3+p_4~4+p_5~4有解,其中x是殆素数P_6,p_j(j=1,…,5)是素数.     

Waring-Goldbach Problem: One Square and Nine Biquadrates

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.     

Density of Integers That Are the Sum of Four Cubes of Primes

gi. IntroductionIt is conjectured that all sufficielltly large integers e satisfying some necessaly congruenceconditions are the sum of four cubes of primes, i.e.(see e.g. [1]). Such a strong conjecture is out of reach at present; but it is reasonable, inview of the following results of Hua and Davenport respectively. A theorem of Hua[3'4] statesthst almost all positive integers, with t * 0, 12(mod 9) are the sum of five cubes of praies,while DavenPOrt's result in 12] asserts that almost all…     

On the Waring-Goldbach Problem for Fourth and Fifth Powers

It is shown that every sufficiently large integer congruentto 14 modulo 240 may be written as the sum of 14 fourth powersof prime numbers, and that every sufficiently large odd integermay be written as the sum of 21 fifth powers of prime numbers.The respective implicit bounds 14 and 21 improve on the previousbounds 15 (following from work of Davenport) and 23 (due toThanigasalam). These conclusions are established through themedium of the Hardy-Littlewood method, the proofs being somewhatnovel in their use of estimates stemming directly from exponentialsums over prime numbers in combination with the linear sieve,rather than the conventional methods which ‘waste’a variable or two by throwing minor arc estimates down to anauxiliary mean value estimate based on variables not restrictedto be prime numbers. In the work on fifth powers, a switchingprinciple is applied to a cognate problem involving almost primesin order to obtain the desired conclusion involving prime numbersalone. 2000 Mathematics Subject Classification: 11P05, 11N36,11L15, 11P55.     

稀疏素数集中的Waring-Goldbach问题(Ⅱ)

本文研究了Waring-Goldbach问题(k=2)与Piatetski-Shapiro素数定理的混合问题,从而进一步深化了华罗庚教授的经典结果.     

On exponential sums over primes and application in Waring-Goldbach problem

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 λsubject 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.     

Exceptional sets in Waring-Goldbach problem for fifth powers

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].     

两个素数平方、四个素数立方和2的整数幂*

本文中, 作者考虑了 Linnik 型的非齐次幂的Waring-Goldbach问题.具体地说, 作者证明了所有充分大的偶数都可以表示成两个素数的平方、四个素数的立方和18个2的正整数幂之和的形式.这改进了Zhao的结果, 即需要43个2的正整数幂.     

On the Waring-Goldbach Problem for One Square and Five Cubes in Short Intervals

Czechoslovak Mathematical Journal - Let N be a sufficiently large integer. We prove that almost all sufficiently large even integers n ∈ [N ? 6U, N + 6U] can be represented as $$left{...     

混合方次的Waring——Goldbach问题

研究混合方次的Waring--Goldbach     

On Waring's Problem for Six Cubes and Higher Degrees

It is proved that the equation $$n = x_{\,1}^3 + x_{\,2}^3 + x_{\,3}^3 + x_{\,4}^3 + x_{\,5}^3 + x_{\,6}^3 + u^4 + v^9$$ has nonnegative integral solutions if $n \equiv 1\left( {\bmod 5} \right)$ is even and sufficiently large. Bibliography: 8 titles.     

Waring-Goldbach Problem for Unlike Powers

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.     

关于多项式的Waring问题（Ⅱ）

设ｆ＿ｌ（ｘ）是首项系数为正的ｌ次整系数多项式，满足条件：不存在整数ｄ，ｑ＞１使得ｆ＿ｌ（ｘ）≡ｄ（ｍｏｄｑ）对所有ｘ成立．记Ｒ＿ｋ（ｎ）为方程的正整数的解数，本文的主要结果是：对于及充分大的ｎ，我们有Ｒ＿ｋ（ｎ）》这是Ｖａｕｇｈａｎ关于Ｗａｒｉｎｇ问题的一个结果对多项式的推广。     

On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)^1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55.     

九个几乎相等的素数的立方之和

本文证明了每个充分大的奇数N.可以表为九个几乎相等的素数的立方之和, 即N=p13+…+p39,这里|Pj-3((N/9)~(1/2))|≤U=N1/3-2/555+ε,从而进一步深化了华罗庚教授的经典结果．我们利用Dirichlet多项式的混合型估计及一个新的迭代方法建立了这一结果．     

Slim exceptional set for sums of two squares,two cubes,and two biquadrates of primes

We prove that, with at most O\left(N^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon }\right) exceptions, all even positive integers up to Nare expressible in the form p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}+p_{\mathrm{3}}^{\mathrm{3}}+p_{\mathrm{4}}^{\mathrm{3}}+p_{\mathrm{5}}^{\mathrm{4}}+p_{\mathrm{6}}^{\mathrm{4}},where p_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}} are prime numbers. This gives large improvement of a recent result O\left(N^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon }\right) due to M. Zhang and J. J. Li.     

On the Multiplicity Problem and the Isomorphism Problem for the Four Subspace Algebra

Let Λ be the four subspace algebra. We show that for any Λ-module M there exists an algorithm (up to the problem of finding roots of the so-called characteristic polynomial of M) with relatively low polynomial complexity of determining multiplicities of all direct summands of M. Moreover, we give a fully algorithmic criterion for deciding if two Λ-modules M and N are isomorphic.     

Waring-Goldbach problems for unlike powers with almost equal variables

We prove that almost all positive even integers n can be written as n=p22+p33+p44+p55 with |pkk-N4|≤N321325+? for 2≤k≤5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N=p1+p22+p33+p44+p55 with |pkk-N5|≤N321325+?for 1≤k≤5.     

A Hybrid of Theorems of Goldbach and Piatetski-Shapiro

It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 p2,p1∈Pγis solvable, with 13/15 <γ≤1, where Pγ= {p| p = [mγ/1], for integer to and prime p} is the set of the Piatetski-Shapiro primes.     

On the Waring-Goldbach problem for seventh powers

We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of seventh powers of prime numbers.