Finding strong pseudoprimes to several bases. II

Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Upper bounds for were first given by Jaeschke, and those for were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

We conjecture that

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.

  

裂纹与线夹杂垂直接触问题的奇性分析

以短纤维复合等作为工程背景，采用现有文献中单裂纹和单夹杂的基本解，对于限平项（基体）上，裂纹和线夹杂的垂直接触问题从断理解力学的角度作了研究，得到了问题的积分方程，推出了接触点的性线指数，奇性应力及以此表示的接触点附近三个区域内的应力强度因子表达式，并给出一些数值结果，可供工程实际参考。  

Real solutions of the equation ϕ(X)=1/nJ n

For ann x n real matrixX, let ϕ(X)=X ο (X ⁻¹) ^T , where ο stands for the Hadamard (entrywise) product. SupposeA, B, C andD aren x n real nonsingular matrices, and among them there are at least one solutions to the equation ϕ(X)=1/nJ _n . An equivalent condition which enable $M = \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right)$ become a real solution to the equation ϕ(X)=1/2nJ _2n , is given. As applications, we get new real solutions to the matrix equation ϕ(X)-1/2nJ _2n by applying the results of Zhang, Yang and Cao [SIAM. J. Matrix Anal. Appl, 21 (1999), pp: 642–645] and Chen [SIAM. J. Matrix Anal. Appl, 22 (2001), pp:965–970]. At the same time, all solutions of the matrix equation ϕ(X)=1/4J ₄ are also given.  

Double penalized variable selection procedure for partially linear models with longitudinal data

Based on the double penalized estimation method,a new variable selection procedure is proposed for partially linear models with longitudinal data.The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components.Under some regularity conditions,the rate of convergence and asymptotic normality of the resulting estimators are established.In addition,to improve efficiency for regression coefficients,the estimation of the working covariance matrix is involved in the proposed iterative algorithm.Some simulation studies are carried out to demonstrate that the proposed method performs well.  

On the Congruence σ(n)≡1(mod n)

Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.  

The structures for the loop-Witt algebra

The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the loop-Witt algebra.  

一类基于进入过程保险风险模型的破产概率(英文)

In this note,one kind of insurance risk models with the policies having multiple validity times are investigated.Explicit expressions for the ruin probabilities are obtained by using the martingale method.As a consequence,the obtained probability serves as an upper bound for the ruin probability of a newly developed entrance processes based risk model.  

一类四阶变系数振动问题的复合解

以一类斜拉索模型为背景,讨论了一类四阶变系数振动问题.用匹配法导出了方程的复合解,从而较简捷地得出了一类斜拉索的振幅的渐近表达式,并为解决相关类型的变系数问题提供了一种有效的方法.  

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.  

Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .