Perspectives de l'Approximation Diophantienne et de la Transcendance

Nous discutons de certains aspects de l'approximation diophantienne et de la transcendance inspirés par deux problèmes encore ouverts: trouver une version effective du théorème de finitude de Siegel pour les points entiers sur les courbes et trouver une approche intrinsèque aux résultats de transcendance des valeurs spéciales des fonctions modulaires sans utiliser les variétés abéliennes (problème posé par Th. Schneider).     

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.     

奇素数的三项方幂和中的平方数

研究了一类基本而又重要的指数Diophantine方程,利用广义Ramanujan-Nagell方程的性质证明了这类方程有非负整数解的充要条件,并得出这类方程的全部非负整数解.     

Geometry and Growth Rate of Frobenius Numbers of Additive Semigroups

Linear combinations x ₁ a ₁ ⋯ x _n a _n of n given natural numbers a _s (with nonnegative integral coefficients x _s ) attain all the integral values, starting from some integer N(a), called the Frobenius number of vector a (provided that the integers a _s have no common divisor, greater than 1). The growth rate of N(a) with the large value of σ = ta ₁ ⋯ a _n depends peculiarly from the direction α of the vector a = σα. The article proves the lower bound of order and the upper bound of order σ ². Both orders are reached from some directions α. The averaging of N(a) along all directions, performed for σ = 7, 19, 41 and 97, provides the values, confirming the rate σ ^p for some p between 3/2 and 2 (for n = 3), excluding neither 3/2 nor 2, for the asymptotic behaviour at large σ. One gets check p ≈ 1, 66 for σ between 100 and 200. These unexpected results, based on some strange relations of the Frobenius numbers to the higher-dimensional continued fractions geometry, lead to many facts of this arithmetic trubulence theory, discussed in this article both as theorems and as conjectures. ^★ Partially supported by RFBR grant 05-01-00104.     

Greek ladders via linear algebra

Theon's ladder is an ancient method for easily approximating nth roots of a real number k. Previous work in this area has focused on modifying Theon's ladder to approximate roots of quadratic polynomials. We extend this work using techniques from linear algebra. We will show that a ladder associated to the quadratic polynomial ax ² + bx + c can be adjusted to approximate either root. Other situations such as quadratics with no real roots and corresponding matrices with complex eigenvalues are also addressed.     

On the ratio of values of a polynomial

For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

直径为4的整树新类

整图是指图的邻接矩阵的特征值全为整数的图. 研究了直径为4的整树.通过求解某些确定的丢番图方程,构造了具有无穷多个这样的整树新类,推广了王力工、李学良和张胜贵发表的文章(见Families of integral trees with diameters 4,6 and 8, it Discrete Applied Mathematics, 2004, 136: 349-362)的一些结论.     

广义Lucas序列与不定方程(Ⅰ)

本文用广义Lucas序列的性质推广了Bender和Herzberg关于ax2＋by2=1,24pn的某些结果,并给出方程的可解性的完整结论,还给出这些结果的一些应用．