数论中切比雪夫不等式的一点补充

利用Wallis不等式,得到素数计算函数的一个下界估计.     

Sums of primes and quadratic linear recurrence sequences

Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.     

新的素数检测方法

提出了一个快速而简单的素数检测方法,它的时间复杂性为O(log^(3+ε)N)这里0<ε≤1,空间复杂性为O(logN),N≡3(mod 4)时,时间复杂性为O(log(3+ε)N)这里0<ε≤1,空间复杂性为O(logN),N≡3(mod 4)时,时间复杂性为O(log^(2+ε)N),是迄今为止最快的多项式算法.     

一个素变数的Diophantine不等式(Ⅲ)

令.本文证明了存在N（c）＞0,对任意实数N＞N（c）,不等式有素数解p1,p2,p3,其中c1为一绝对正常数．     

一种快速的素数生成和检验算法

就运算复杂度、报错率、实际运行效率等方面,对已有的素数检验算法进行了分析和比较.同时分析素数生成的相关算法,优化了ISO/IEC的生成算法并得到一个新的素数生成算法.     

具有弱$s$-置换嵌入子群的有限群

Let P be a Sylow p-subgroup of a group G with the smallest generator number d,where p is a prime.Denote by M_d(P) = {P_1,P_(2,...,)P_d} a set of maximal subgroups of P such that φ(P) = ∩_(n=1)~dP_n.In this paper,we investigate the structure of a finite group G under the assumption that the maximal subgroups in M_d(P) are weakly s-permutably embedded in G,some interesting results are obtained which generalize some recent results.Finally,we give some further results in terms of weakly s-permutably embedded subgroups.     

与ω函数的动力学相关的逆问题

用P记素数的集合,P（n）表示整数n（〉1）的最大素因子．记C3={p1p2p3：Pi∈P（i=1,2,3）,pi≠pj（i≠j）}, B3={p1p2p3：pi∈P（i=1,2,3）,p1=p2或p1=p3或p2=p3,但非p1=p2=p3}． 对于礼=p1p2p3∈C3UB3,定义训函数为 w（n）=P（p1＋p2）P（p1＋p3）P（p2＋p3）．如果有m∈S∪→C3∪B3,使得w（m）=n,则称m为n的S-亲源． 本文证明：在C3中有无穷多个元,它们有足够多的C3-亲源;在B3中有无穷多个元,它们也有足够多的C3-亲源．本文还证明,在B3中有无穷多个元,它们有足够多的B3-亲源．     

Finite groups with its power automorphism groups having small indices

In this paper, a finite group G with IAut（G） ： P（G）I ~- p or pq is determined, where P（G） is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies ｜Aut（G） ： P（G）｜= pq if and only if Aut（G）/P（G） ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.     

Sums of nine almost equal prime cubes

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.