不定方程解的δ-组合计数方法及应用

运用徐利治和 Ｐｅｔｅｒ Ｊａｕ－Ｓｈｙｏｎｇ Ｓｈｉｕｅ的 Ｋｒｏｎｅｃｋｅｒ δ符号的组合计数方法,可以给出一般算术方程ｆ（Ｘ）＝ｍ在给定区域内的解之计数公式,由此可产生无穷多个组合恒等式,并能引出一些不等式．     

非线性不定方程的整数解

不定方程是数论中的一个课题,它的整数解问题是国内外数学竞赛的一种命题,常见的有两类:一类是需要求方程的整数解。另一类是证明方程无整数解。由于一次不定方程(组)问题许多书刊都作些探讨,这里不再赘述。本文仅对数学竞赛(特别是一些级别较高的竞赛)中常见的非线     

n元一次不定方程通解的特征及应用

ｎ元一次不定方程通解的特征及应用叶萍恺（浙江省少数民族师范学校３２３０００）ｎ元一次不定方程ｂ１ｘ１＋ｂ１ｘ２＋…＋ｂｎｘｎ＝ｄ＝１（１）（其中。２２，ｈｉ，６２，…，ｂ。都是非零整数，且（ｈｉ，ｂＺ，…，ｂ。）＝１，ｄ是整数，在此先考虑ｄ—１的情形...     

一类不定方程恒有正整数解的几个判别法则

一类不定方程恒有正整数解的几个判别法则邱天绪，树华（老河口师专４４１８００）《数学通讯》１９９５年第１期，在问题征解栏中，有如下一道数学征解题：“证明对于任意自然数ｎ，方程ｘ２＋２ｙ２＝１７ｎ（Ⅰ）恒有整数解．作者发现此不定方程尚可进一步的推广．本文...     

不定方程6y^2=x（x＋1）（2x＋1）的解的简洁初等证明

不定方程６ｙ￣２＝ｘ（ｘ＋１）（２ｘ＋１）的解的简洁初等证明汉江机床厂何宗友１８７５年卡斯（Ｌｕｃａｓ）问不定方程或是否仅有非平凡解ｘ＝２４，ｙ＝７０．１９１９年沃森（Ｗａｔｓｏｎ）、１９５２年琼格伦（Ｌｊｕｎｇｇｒｅｎ）分别利用椭圆函数与二次域理论...     

W．J．LeVeque的一个定理的注记

本文证明了：对固定的正整数α，β．ｍ，其中ｍ≥２，若方程　有无穷多个正整数解ｎ，则ｍ＝２，α＝３及β＝１．这推广了ＬｅＶｅｑｕｅ的一个结果。     

15个著名的不定方程问题

１５个著名的不定方程问题杨仕椿（四川西充县常林中学６３７２００）不定方程的研究，是数论研究中的热点与难点，它以其内容简洁易懂而方法艰深困难向人类的智慧提出挑战．除了著名的Ｆｅｒ－ｍａｔ大定理外，还有许多的其它不定方程问题，本文列举出１５个问题，逐一阐...     

具有四项的指数丢番图方程(Ⅲ)

根据定理 １,２和 ３;求任何一个方程 ａ~ｘ－ｂ~ｙ＝ｎ,ａ~ｘｂ~ｙ±ａ~ｚ±ｂ~ｗ±１＝０ 或ａ~ｘ±ｂ~ｙ±ａ~ｚ±ｂ~ｗ＝０（ｘ,ｙ,ｚ,ｗ∈≥０）的解都是很简单的,此处ａ,ｂ是适合 ２ ≤５ ａ,ｂ≤５０的互素的两个整数,ｎ是适合１≤ｎ≤８００００的整数．     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

On Some Exponential Diophantine Equations

 We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000)     

关于指数型丢番图方程的整数解

本文简要地介绍了有关Ｓ－单位方程、Ｒａｍａｎｕｊａｎ－Ｎａｇｅｌｌ方程、Ｔｈｕｅ－Ｍａｈｌｅｒ方程、ＬｅＶｅｑｕｅ方程、Ｃａｔａｌａｎ方程、Ｐｉｌｌａｉ方程等指数型丢番图方程整数解的最新结果。这些结果大多是用Ｔｈｕｅ－Ｓｉｅｇｌｅ－Ｒｏｔｈ－Ｓｃｈｍｉｄｔ方法和Ｇｅｌ’ｆｏｎｄ－Ｂａｋｅｒ方法得到的。     

一类非线性算子方程的多重正解及其应用

本文利用不动点指数理论对一类非线性算子方程建立了多重正解的存在性定理，并将所获结果应用到Ｈａｍｍｅｒｓｔｅｉｎ积分方程和Ｓｔｕｒｍ Ｌｉｏｕｖｉｌｅ两点边值问题，得到了新的结论，本质上改进和推广了［２，４，５，６，８－１２］的工作．     

丢番图方程及Browkin的一个猜想

本文首先证明了一些丢番图方程在整数环Z中无解,然后我们部分验证了Browkin在[Lecture Notes in Math.,Vol.966,1-6]中的一个猜想.     

一类丢番图方程与有限域上对角方程的解

给出了一类丢番图方程的解数为11,…,18时,其最小整数解的具体表达式,并推广得到该类丢番图方程的解数为素数p时,其最小整数解的具体表达式.还补充了该类丢番图方程的解数为6,8,10时,w的具体值,其中w为有限域F_q上简单对角方程的次数向量d=(d₁,…,d_n)的压缩向量.     

一类非线性方程的解的存在性及其应用

设Ａ是Ａｍａｎｎ意义下的凹（凸）算子,本文提出序Ｌｉｐｓｃｈｉｔｚ条件,无需考虑任何紧性或连续性条件,由Ｍａｎｎ迭代技巧证明了方程Ａｘ＝ｘ的解的存在性,将所得结果应用于无辊域ａｍｍｅｒｓｔｅｉｎ发方程,得到了新结果。     

Solving Some Quadratic Diophantine Equations with Clifford Algebra

In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type ${{a_1x_1^2+ .,.,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .,.,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .,.,.,,p+q,x_{n+1}neq0)}}${{a_1x_1^2+ .,.,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .,.,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .,.,.,,p+q,x_{n+1}neq0)}} are found using simple reflections of orthogonal vectors, manipulated using the Clifford algebra over orthogonal spaces R ^p,q . These solutions are obtained from the application of a useful Lemma: given two different non-zero vectors of the same norm, we can map one onto the other, or its negative, by means of a simple reflection. With this Lemma, we extend and improve a previous work [1] concerning generalized Pythagorean numbers, which now can be obtained as a Corollary. We also show that our technique is promising for solving others Diophantine equations.