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

设r是大于1的正奇数,m是正偶数,V(r)+U(r)(-1)~(1/2)=(m+(-1)~(1/2))~r.本文证明了:当a=|V(r)|,b=|U(r)|,c=m~2+1时,如果r≡5(mod8),m>r~2且r<11500或者m>2r/π且r>11500,则方程a~x+b~y=c~z仅有正整数解(x,y,z)=(2,2,r).     

Mordell方程y~3=x~2+2p~4的正整数解

设p是奇素数.对于非负整数r,设U_(2r+1)=(α~(2r+1)+β~(2r+1))/2~(1/2),V_(2r+1)=(α~(2r+1)-β~(2r+1))/6~(1/2),其中α=(1+3~(1/2))/2~(1/2),β=(1-3~(1/2))/2~(1/2).运用初等数论方法证明了:方程y~3=x~2+2p~4有适合gcd(x,y)=1的正整数解(x,y)的充要条件是p=U_(2m+1),其中m是正整数.当上述条件成立时,方程仅有正整数解(x,y)=(V(2m+1)(V_(2m+1)~2-6),V_(2m+1)~2+2)适合gcd(x,y)=1.由此可知:当p10000时,方程仅有正整数解(p,x,y)=(5,9,11),(19,1265,123),(71,68675,1683)和(3691,9677201305,4541163)适合gcd(x,y)=1.     

Fermat数的Smarandache函数值的下界

对于正整数n,设F_n=2~2~n+1是第n个Fermat数,又设S(F_n)是F_n的Smarandache函数.运用初等的方法证明了:当n≥4时S(F_n)≥4·2~n(4n+9)+1.     

一个素变量丢番图问题

设k和r是满足k≥3及r≥Ψ(k)+1的正整数,这里当3≤k≤4时,Ψ(k)=2~(k-1);而当k≥5时,Ψ(k)=1/2k(k+1).假定δ和ε是给定的足够小的正数,λ_1,λ_2,…,λ_(r+1)是不全同号且两两之比不全为有理数的非零实数.对于任意实数η与0σ2~(1-2k)/r-1,证明了:存在一个正数序列X→+∞,使得不等式|λ_1p_1~k+λ_2p_2~k+···+λ_rp_r~k+λ_(r+1)p_(r+1)+η|(max(1≤j≤r+1)p_j)~(-σ)有》■X~(■-(2~(1-2k))/(r-1)+ε组素数解(p_1,p_2,…,p_(r+1)),这里(δX)~(1/k)≤p_j≤X~(1/k)(1≤j≤r)及δX≤p_(r+1)≤X.这改进了之前的结果.     

问题与解答

一本期问题 1.设a、m、n是正整数,n是奇数。证明数a~n-1和a~m+1的最大公因数不大于2。 2.证明数2~(5n+1)+5~(n+2)当n=0,1,2,…时,可以被27整除。 3.求出一个形如的整数的九位数,此数是四个不同素数的平方之积,且,(a_1≠0)。     

一个代数不等式的加强

近两年,在众多刊物上,载有不等式: multiply from i=1 to n(x_i+1/x_i)≥(λ/n+n/λ) (*)这里x_i∈R~+(i=1,2,…,n),x_1+x_2+…+x_n=λ≤n,仅当x_1=x_2=…=x_n时(*)式取等号。现在,我们给出(*)的一个加强: 定理设x_i∈R~+(i=1,2,…,n,n≥2),且sum from i=1 to n x_i=λ(常数)≤n,则 sum from i=1 to n(x_i+1/x_i)~(-1)≤n(λ/n+n/λ)~(-1) (1)当且仅当x_1=x_2+…=x_n时,(1)式中的等号成立。     

整数环上一类矩阵方程Xn+Yn=λnI(n∈N,λ∈Z,λ≠0)的解

设Z,N分别是全体整数和正整数的集合,M_m(Z)表示Z上m阶方阵的集合.本文运用Fermat大定理的结果证明了:对于取定的次数n∈N,n≥3,二阶矩阵方程Xⁿ+Yⁿ=λⁿI(λ∈Z,λ≠0,X,Y∈M₂(Z),且X有一个特征值为有理数)只有平凡解;利用本原素因子的结果得到二阶矩阵方程Xⁿ+Yⁿ=(±1)ⁿI(n∈N,n≥3,X,Y∈M₂(Z))有非平凡解当且仅当n=4或gcd(n,6)=1且给出了全部非平凡解;通过构造整数矩阵的方法,证明了下面的矩阵方程有无穷多组非平凡解:■n∈N,Xⁿ+Yⁿ=λⁿI(λ∈Z,λ≠0,X,Y∈M_n(Z));X³+Y³=λ³I(λ∈Z,λ≠0,m∈N,m≥2,X,Y∈M_m(Z)).     

实二次城 Q(■)类数的可除性

设 d 是无平方因子正整数,h(d)是实二次域 Q(d~(1/2))的类数.本文证明了:如果 da~2=1+4k~(2n),a、k、n 是正整数,k>1,n>1,n 的奇素因子 p和 k 的素因子 q 都适合 gcd(p,(q-1)q)=1,而且 2k~n+ad~(1/2)是 Pell 方程u′~2-dv′~2=-1 的基本解,则除了(a,d,k,n)=(5,41,2,4) 以及 n=2,k=P_mP_(m+1) 或者 2Q_mQ_(m+1) 以外,h(d)=0(modn),这里 m 是正整数,P_m=1/2((1+2~(1/2))~m+(1-2~(1/2))~m),Q_m=1/22~(1/2)((1+2~(1/2))~m-(1-2~(1/2))~m).由此可推得:对于任何正整数 n,存在无限多个实二次域,可使 n 整除其类数.     

实二次城 Q(■)类数的可除性

设 d 是无平方因子正整数,h(d)是实二次域 Q(d~(1/2))的类数.本文证明了:如果 da~2=1+4k~(2n),a、k、n 是正整数,k>1,n>1,n 的奇素因子 p和 k 的素因子 q 都适合 gcd(p,(q-1)q)=1,而且 2k~n+ad~(1/2)是 Pell 方程u′~2-dv′~2=-1 的基本解,则除了(a,d,k,n)=(5,41,2,4) 以及 n=2,k=P_mP_(m+1) 或者 2Q_mQ_(m+1) 以外,h(d)=0(modn),这里 m 是正整数,P_m=1/2((1+2~(1/2))~m+(1-2~(1/2))~m),Q_m=1/22~(1/2)((1+2~(1/2))~m-(1-2~(1/2))~m).由此可推得:对于任何正整数 n,存在无限多个实二次域,可使 n 整除其类数.     

Fermat素数与Jemanowicz猜想

设r是正整数.本文运用初等数论方法证明了方程((2~(r+1)+1)n)~x+((2~(2r+1)+2~(r+1)n)~y=((2~(2r+1)+2~(r+1)+1)n)~2适合(x,y,z)≠(2,2,2)以及n1的正整数解(x,y,z,n)都满足xzy特别是当2~(r+1)是素数时,该方程仅有正整数解(x,y,z,n)=(2,2,2,t),其中t是任意正整数,即此时Jemanowicz猜想成立.     

SR分解的乘法扰动界

In this paper, we present the first order perturbation bounds for the SR factorization with respect to left multiplicative perturbation, and the first order and rigorous perturbation bounds for this factorization with respect to right multiplicative perturbation.Moreover, taking the properties of SR factors into consideration, we also provide some refined perturbation bounds.     

Hermite矩阵特征值的新扰动界

本文研究了Hermite矩阵特征值的任意扰动,给出了新的绝对和相对扰动界．所给出的界改进了Hoffman-Wielandt和Kahan早期的结果．     

The Numbers of Shared Upper Bounds Determine a Poset

We prove that a finite poset P=(V,≤) is determined up to some permutation of its elements by the function defined by m_P({u,v})=|{w∈V∣u≤w and v≤w}|.     

Bounds of deviation for branching chains in random environments

We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the process.     

On Uniform Global Error Bounds for Convex Inequalities

This paper studies the existence of a uniform global error bound when a convex inequality g 0, where g is a closed proper convex function, is perturbed. The perturbation neighborhoods are defined by small arbitrary perturbations of the epigraph of its conjugate function. Under certain conditions, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if g satisfies the Slater condition and the solution set is bounded or its recession function satisfies the Slater condition. The results are used to derive lower bounds on the distance to ill-posedness.     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

Bounds for the sum of distances of spherical sets of small size

We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when $N=\mathrm{\Theta }(n^{\alpha }),0<\alpha ?2$. The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.     

On Lower Bounds For Covering Codes

We study lower bounds on K(n,R), the minimum number of codewords of any binary code of length n such that the Hamming spheres of radius R with center at codewords cover the Hamming space . We generalize Honkala's idea toobtain further improvements only by using some simple observationsof Zhang's result. This leads to nineteen improvements of thelower bound on K(n,R) within the range of .     

Sharp Upper and Lower Bounds for Basket Options

Given a basket option on two or more assets in a one‐period static hedging setting, the paper considers the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. Sharp upper bounds are derived for the general n‐asset case and sharp lower bounds for the two‐asset case, both in closed forms, of the price of the basket option. In the case n = 2 examples are given of discrete distributions attaining the bounds. Hedge ratios are also derived for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves.     

关于实二次域类数的上下界

设Ｄ是无平方因子正整数，Δ，ｈ，ε分别是实二次的判别式、类数和基本单位数，本文运用初等方法证明了：（ｉ）在假定广义Ｒｉｅｍａｎｎ猜想成立的条件下，当Δ＞１０￣９时，（ｉｉ）当Ｄ＝ｐ，ｐ是奇素数时，