数学问题解答北大核心

2017年12月号问题解答（解答由问题提供人给出）2396形如n=16a（16b+15）（a,b∈N）的正整数不能表示成14个整数的四次方和.（浙江省富阳二中许康华311400）证明由x4≡0,1（mod16）,得对任意的x1,x2,…,x14∈Z,都有x14+x24+…+x144≡0,1,2,…,14（mod16）假设当a=l∈N,b∈N结论都成立.当a=l+1时,如果存在某个b∈N。     

奇完全数的两个猜想

运用初等方法讨论有关奇完全数的两个猜想.证明了:(i)如果n=p~αq_1~(2β_1)q_2~(2β_2)…q_s~(2β_s)是奇完全数,其中P,q_1,q_2,…,q_s是不同的奇素数,α,β_1,β_2,…,β_s是正整数,p≡α≡1(mood4),而且q_i≡-1(mod m)(i=1,2,…,s),m是大于2的正整数,则.1/2σ(p~α)必为合数;(ii)如果n=a~2~x+b~2~x,其中a,b,x是适合ab,gcd(a,6)=1,2|ab的正整数,则当x≥log_2log_2log_2 a时,n不是奇完全数.     

除数为11的速算法

由欧拉定理,费尔马小定理可知,既约真分数a/b可化为纯循环小数的充分必要条件是(b,10)=1,循环节长度是使10~t≡1(mod b)成立的最小正整数t。当b=11,(b,10)=1,10~2≡1(mod 11)。循环节长度为2。当除数为11时。可利用11的特殊性进行速算。     

关于本原商高数的新猜想

设a,b,c是满足a=m²-n2-n²,b=2mn,c=m2,b=2mn,c=m²+n2+n²的正整数,其中m,n是适合m>n,gcd(m,n)=1,2|mn的正整数.运用初等数论方法讨论了方程c2的正整数,其中m,n是适合m>n,gcd(m,n)=1,2|mn的正整数.运用初等数论方法讨论了方程c^x+bx+b^y=ay=a^z的正整数解(x,y,z).证明(m,n)≡(0,1),(0,5),(1,2),(2,3),(3,4),(4,1),(4,5),(5,6),(6,7)或(7,0)(mod8)时,方程无解.上述结果部分地解决了有关本原商高数的一个新猜想.     

A Conjecture Concerning the Pure Exponential Diophantine Equation a^x ＋ b^y = c^z

Let a, b, c, r be fixed positive integers such that a^2 ＋ b^2 = c^r, min（a, b, c, r） 〉 1 and 2 r. In this paper we prove that if a ≡ 2 （mod 4）, b ≡ 3 （mod 4）, c 〉 3.10^37 and r 〉 7200, then the equation a^x ＋ b^y = c^z only has the solution （x, y, z） = （2, 2, r）.     

On the Terai-Jésmanowicz conjecture

In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）.
Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.     

关于丢番图方程x~3±1=3·2~αpD_1y~2

设D_1=multiply from i=1 to s q_i(s=1或2),q_i≡-1(mod6)(i=1,2,…,s)是彼此不同的奇素数,p≡1(mod6)为奇素数.运用初等方法讨论了丢番图方程x~3±1=3·2~αpD_1y~2(α=0或1)的正整数解的情况.     

关于D.H.Lehmer问题与超级Kloosterman和

设整数q>2,c与q互素．对于1到q之间与q互素的任意整数a,在1到q之间存在唯一的整数b满足ab≡c mod q．对任意整数k≥2,定义M(q,k,c)为满足1≤ai≤q, (ai,q)=1,i=1,2,…,k,a1a2…ak≡c mod q且2 a1 a2 … ak的正整数组(a1,a2,…,ak)的数目,并设E(q,k,c)=M(q,k,c)-(φk-1(q))/2．本文的主要目的是利用Gauss和与原特征的性质,以及Dirichlet L-函数的均值定理,来研究E(q,k,c)与超级Kloosterman和K(h,k,q)的混合均值,并给出一个均值公式．     

关于丢番图方程X~2+4Y~4=pZ~4

设p为素数,p=4A~2+1+2|A,A∈N~*.运用二次和四次丢番图方程的结果证明了方程G:X~2+4Y~4=pZ~4,gcd(X,Y,Z)=1,除开正整数解(X,Y,Z)=(1,A,1)外,当A≡1(mod4)时,至多还有正整数解(X,Y,Z)满足X=|p(a~2-b~2)~2-4(A(a~2-b~2)±ab)~2|,Y~2=A(a~2-b~2)~2±2ab(a~2-b~2)-4a~2b~2A,Z=a~2+b~2;当A≡3(mod4)时,至多还有正整数解(X,Y,Z)满足X=|4a~2b~2A-(4abA±(a~2-b~2))~2|,Y~2=4a~2b~2A±2ab(a~2-b~2)-A(a~2-b~2)~2,Z=a~2+b~2.这里a,b∈N~*并且ab,gcd(a,b)=1,2|(a+b).同时具体给出了p=5时方程G的全部正整数解.     

Diophantine方程组a~2+b~2=c~r和x~2+b~y=c~z的一点注记

设r是大于1的正奇数,a,b,c是满足a~2+b~2=c~r的互素正整数.证明了:当r(?)5(mod8),c>10~(12)r~4且b是奇素数的方幂时,方程x~2+b~y=c~z仅有正整数解(x,y,z)=(a,2,r).     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.     

THE 8~(th) INTERNATIONAL CONGRESS ON INDUSTRIAL AND APPLIED MATHEMATICS

<正>August 10-14,2015Beijing,ChinaThe International Congress on Industrial and Applied Mathematics(ICIAM)is the premier international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics.From August 10 to 14,2015,mathematicians,scientists     

Towards Excellence in Science Chinese Academy of Sciences

<正>May 26,2014,Beijing Science is a human enterprise in the pursuit of knowledge.The scientific revolution that occurred in the 17th Century initiated the advances of modern science.The scientific knowledge system created by     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.