平方数

The sequence 1, 4, 9, 16,… lists all positive square numbers in ascending order.

数列 1、4、9、16、…把所有正平方数由小至大排列。

In 2000, F. LUCA proved that Fermat number are anti-sociable numbers, and in 2005, M. H. LE proved all powers of 2 are anti-sociable numbers. We have used the method of M. H. LE to obtain some new results of the anti-sociable numbers. For every integer n containing prime divisors that are 1 mod 4, let p mod 4 be an arbitrary prime divisor of n. There is at least one anti-sociable number in n^2, p^2n^2, p^4n^2, and p^6n^2. Therefore we can prove that anti-sociable numbers have positive density in perfect square numbers. We also give a method to find the exact anti-sociable numbers.

LUCA证明了Fermat数都是孤立数；2005年，乐茂华教授证明了2的方幂都是孤立数，用乐茂华教授的方法给出孤立数的一些新的结果：对于任意含有4w+1型素因子的正整数n，设p为n的任意一个4w+1型素因子，则在n^2，p^2n^2，p^4n^2，p^6n^2里至少有一个是孤立数，因此可以证明孤立数在完全平方数里有正密度，另外也给出求解确定孤立数的方法。

In celebration of the Company supplies on a rich resources, there are nearly 5,000 square feet of new red, blue carpet; several different colors of the 10 arches; cut a full set of gold; hundreds of square dedicated board; thousands of the colorful flags; few 100-meter red in the Buman…… Acting resources: a balloon launch, dragon and lion dance teams, Yaogu team and the military band, a bike speed performance, we have a good understanding in the gift area also has abundant Resources: all kinds of pop, the new, practical gifts.

本公司在庆典用品上有着丰富的资源，有近5000平方的全新红、蓝地毯；数十条颜色不一的拱门；金剪全套；数百平方的专用板；数千面的彩旗；数百米的红金布幔……在代理资源上：升空大气球、舞龙舞狮队、腰鼓队、军乐队、自行车极速表演，我们均有良好的默契；在礼品方面上也有着丰富的资源：各式流行，新款，实用礼品。

In this paper we introduce a search algorithm of computer, and the arithmetic can get a magic square of squares with high rank in a short time.

本文提供了一种计算机搜索算法，该算法能够在比较短的时间内获得阶数较大的平方数幻方。

The existence of global weak solution for a class of generalized Ginzburg-Landau equations with an inhomogeneous boundary condition was studied. Some integral indentities of smooth solution of inhomogeneous initial boundary value problem of Ginzburg-Landau equations were deduced, by which a priori estimates of the square norm on boundary of normal derivative and the square norm of partial derivatives were obtained.

研究具非线性边界条件的一类广义Ginzburg＿Landau方程解的整体存在性·推导了Ginzburg＿Landau方程的非齐次初边值问题光滑解的几个积分恒等式，由此得到了解的法向导数在边界上的平方模以及解的平方模和导数的平方模估计；通过逼近技巧、先验估计和取极限方法证明了Ginzburg＿Landau方程的非齐次初边值问题整体弱解的存在性

For all integers , is a perfect square.

对於每个整数，都是完全平方数。

How many of the sets S0,S1,S2,……,S999 do not contain a perfect square?

在集合S0，S1，S2，……，S999中有多少个不包括完全平方数？

In addition, by utilizing Jacobi two square numbers theorem and Lagrange four square numbers theorem and some theta function identities, we also prove the known results of number theory: two triangular number theorem, four triangular number theorem, and the number of representations of a positive integer by various quadratic forms in terms of divisor functions

包括Jacobi二平方数定理，Lagrange四平方数定理等，然后利用这些结果结合几个theta函数恒等式，我们获得了把任意一个正整数表示成两个三角数或四个三角数的和以及其他的二次形式的方法数，这些方法数都是用因子函数来表示的。

This is basic andit has many applications in the theory of diophantine equations.

并证明了，该D为给定正整数，〓且非平方数，则〓〓，其中k、l过所有适合kl=D的正整数，中有且仅有一个方程有解，这一在不定方程中有许多应用的基本结论。

Prove that if n is not a square then it has an even numble of divisors.

证明：若n不是完全平方数，则n有偶数个因数。

Neither the killing of Mr Zarqawi nor any breakthrough on the political front will stop the insurgency and the fratricidal murders in their tracks.

在对危险的南部地区访问时，他斥责什叶派民兵领导人对中央集权的挑衅行为。

In fact,I've got him on the satellite mobile right now.

实际上 我们已接通卫星可视电话了