一个素变数的Diophantine不等式

设１＜Ｃ＜１３／１２．本文证明了存在Ｎ（Ｃ）＞０使得对任意实数Ｎ＞Ｎ（Ｃ），下面的不等式｜＋＋　－Ｎ｜＜Ｎ－　　　　）ｌｏｇｓＮ有素数解ｐ１，ｐ２，ｐ３，其中ｓ＝２（１５）．     

三个或多个素变数的线性方程

推广了J.B.Friedlander和D.A.Goldston的结果,给出了素变数整系数线性方程a1p1 a2p2 … akpk=N(k≥3)解的个数的渐近公式.     

小区间上的三素数定理

本文给出了素变数方程:N=p_1+p_2+P_3,N/3-U相似文献   

小区间上的素变数三角和

建立了小区间上的素变数三角和的一个新估计, 利用这一估计, 证明了每个充分大的模24同余于5的整数N可以表为 这里p_j 是素数. 这个无条件结果深化了华罗庚五素数平方定理, 而且其质量与以往在广义Riemann假设下所得的结果相同.     

素变数三角和的估计及其在Waring-Goldbach问题中的应用

证明了关于素变数三角和的如下估计: 设k≥1, , x≥2, 满足(a, q) = 1, 1≤a≤q, 和λ∈R, 则 作为应用, 证明了: 除了至多个例外, 所有满足必要条件的正整数n≤N都是三个素数的平方和. 这一结果与前人在广义Riemann假设之下所得结果一致.     

素变数非线性型的整数部分

证明了:假设λ,μ是不全为负的非零实数,λ是无理数,k是正整数,那么存在无穷多素数p,p_1,p_2,使得[λp_1+μp_2~2]=kp.特别地,[λp_1+μp_2~2]表示无穷多素数.     

一个素数,两个素数的平方以及2的若干次幂和的丢番图逼近

在给定条件下证明了不等式|λ1p1 λ2p22 λ3p23 μ12x1 … μs2xs γ|＜η有无限多素数p1,p2,p3和正整数x1,…,xs解.     

一目到底加基弃5加减法

一目到底加减法属于脑珠结合速算，其算法很多，大致可归为三大类，一是心算直加直减类，主要有累计滚加法、凑整加法，记“5”加法和加减抵销法；二是加基抛5类，主要有加基抛5加法、加基抛5变数加减法和按行加基排5计尾法；三是加补减齐类，主要有加补减齐加减法。     

关于多复变数Cauchy型积分的边界性质

本文用新的方法研究Ｂ－Ｍ型积分的边界性质，所得结果推进了文［１］的结果，并指出文［４］证明有错误     

混合幂的素变数丢番图逼近

证明了:如果λ_1,λ_2,λ_3,λ_4是正实数,λ_1/λ_2是无理数和代数数,V是well-spaced序列,δ0,那么对于v∈V,v≤X,ε0,使得|λ_(1p_1~2)+λ_(2p_2~2)+λ_(3p_3~3)+λ_(4p_4~3)-v|v~(-δ)没有素数解p1,p2,p3,p4的v的个数不超过O(X~(20/21+21δ+ε)).     

Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers

In this paper, we prove that there is an arithmetic progression of positive odd numbers for each term of which none of five consecutive odd numbers and can be expressed in the form , where is a prime and are nonnegative integers.

     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

More on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This paper extends these results to show that .

     

New techniques for bounds on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This was extended by the author to show that . Using an idea of Carl Pomerance this paper extends these results. The current new bound is .

     

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

Five consecutive positive odd numbers none of which can be expressed as a sum of two prime powers II

In this paper, we find two integers k0, m of 159 decimal digits such that if k ≡ k0 （mod m）, then none of five consecutive odd numbers k, k - 2, k - 4, k - 6 and k - 8 can be expressed in the form 2^n ± p^α, where p is a prime and n, α are nonnegative integers.     

关于π(x+y)≤π(x)+π(y)的一些研究结果

§ 1.　Introduction　　Indistributionofprimenumbersthereisamorecommonconjecturethanthatof“twinprimenumbers” :Ifdispelledbycongruencerelation ,thereisanunlimitednumberofprimenumbercon coursesofanygiventype .Forexample ,itisreasonabletobelievethatthereisanunlimitednumberofprimenumberconcoursescomposedofthreeasagroup ( {6k- 1 ,6k + 1 ,6k+ 5 }and {6k+ 1 ,6k+ 5 ,6k+ 7}) ,whichcanbenamedas“threeprimenumbers” .Isthereanunlimitednumberof“n primenumbers”suchas {p +k1 ,p +k2 ,… ,p+kn-1 },whichare…     

The odd-number sequence: squares and sums

Direct study of various characteristics of integers and their interactions is readily accessible to undergraduate students. Integers obviously fall in different classes of modular rings and thus have features unique to that class which can result in a variety of formations, particularly with sums of squares. The sum of the first n odd numbers is itself the square of n within the odd number sequence, from which testing for primality within the Fibonacci sequence is investigated in this note.     

Finding prime pairs with particular gaps

By a prime gap of size , we mean that there are primes and such that the numbers between and are all composite. It is widely believed that infinitely many prime gaps of size exist for all even integers . However, it had not previously been known whether a prime gap of size existed. The objective of this article was to be the first to find a prime gap of size , by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from to , and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form , , and their application to divisibility of binomial coefficients by a square will also be discussed.