New results in equal sums of like powers

This paper reports on new results for the equation

i.e., equal sums of like powers. Since the 1967 Lander, Parkin and Selfridge survey paper [4], few other numeric results have been published (see Elkies [6] and Ekl [3]). The present paper reports on several new smallest primitive solutions. Further, search limits have been extended in many cases, and tables of solutions are presented. Additionally, new solutions to the same class of problems in distinct integers have been discovered.

     

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．     

整图的构造

In this paper, some new classes of integral graphs are given in two new ways. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving diophantine equations. Some classes are infinite. The discovery of these classes is a new contribution to the search of such integral graphs.     

1+2x5y+5z11u=2v·11w,xuvw>0,y+z>0的非负整数解

借助计算机,用初等方法给出了指数丢番图方程1+2^x5^y+5^z11^u=2^v·11^w,xuvw>0,y+z>0的全部非负整数解.     

关于指数丢番图方程a~x+(3a~2-1)~y=(4a~2-1)~z的正整数解(英文)

本文通过计算Jacobi符号,运用代数数的对数线性型的下界估计,证明了:当整数a>1时,指数丢番图方程a~x+(3a~2-1)~y=(4a~2-1)~z仅有正整数解(x,y,z)=(2,1,1).     

方程y~2=nx(x~2±1)的正整数解数的上界(英文)

设n是无平方因子正整数.本文利用二次和四次Diophantine方程解数的结果,讨论了方程y~2=nx(x~2±1)的正整数解个数的上界,证明了该方程至多有2~w(n)个正整数解(x,y),其中w(n)是n的不同素因数的个数.     

Diophantine方程nx~2+2~m=y~n(英文)

设n是大于3的奇数.本文运用Y.Bilu,G.Hanrot和P.M.Voutier关于Lehmer数本原素因子存在性的新近结果,证明了方程nx~2+2~m=y~n没有适合gcd(x,y)=1且m为奇数的正整数解(x,y,m).     

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

矩阵乘幂的一种显式公式

本文给出用线性Ｄｉｏｐｈａｎｔｕｓ方程的解计算矩阵乘幂的显式公式，并给出常系数联立线性递推关系组的解的显式表达式。     

Diophantine m-tuples for linear polynomials

In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. This revised version was published online in June 2006 with corrections to the Cover Date.