平方数

平方数是一种很“完美”的数，有关于它有许多很有序的规律。

Square number is a " perfect " number which has many very orderly rules .

对于每个整数，都是完全平方数。

For all integers , is a perfect square .

该例声明一个名为square的子例程，计算某个数字的平方数并返回该数字。

This example declares a subroutine called square that squares the number and returns it .

Fibonacci数列和Lucas数列中的殆平方数

Nearly Square Numbers in the Fibonacci and Lucas Sequences

一类序列的平方数

The Square Numbers of a Class of Sequence

全平方数集中的除数问题

On certain divisor sum over square-full integers

分圆数a-bx中的平方数

The Squares in the Cyclotomic Numbers abx

完全平方数的分布

The distribution of square - full integers

关于调和数中的平方数

On the squares in harmonic numbers

我们知道程序是用来求平方数的，那么按理说我们可以来求-4的平方，也就是16，但是程序结果是怎么样的呢？

We knew this was trying to do squaring , so intellectually we know we can square-4 , it ought to be16 , but what happens here ?

第一，数学是完备的吗？是不是每个命题（比如『任意自然数都是四个平方数的和』）都能证明或证伪。

First , was mathematics complete , in the technical sense that every statement ( such as ' every integer is the sum of four squares ' ) could either be proved , or disproved .

另外，那些已经得到解决的难题，比如四平方数定理，极少有被“机械的规则”证明的，往往都是通过创造性推演，或者构建新的抽象代数概念。

Furthermore , the difficult problems which had been solved , such as Gauss 's Four Square Theorem , had rarely been proved by anything like a ' mechanical set of rules ' , but by the exercise of creative imagination , constructing new abstract algebraic concepts .

比如说，有的算术定理需要无数步来证明――比如拉格朗日定理：任一自然数都是四个平方数的和。谁能保证说，一直往下找，不会遇到矛盾？

There were , for instance , theorems in arithmetic which took thousands of steps to prove - such as Gauss 's theorem that every integer could be expressed as the sum of four squares . How could anyone know for sure that there was not some equally long sequence of deductions which led to a contradictory result ?

一类无平方因子数的k次方和

The Sum of the kth Power of One Type of Square Free Number

的2-有向路色数，x（G~2）表示图G的平方色数。

Let x ( G ~ 2 ) denote the chromatic number of square of a graph G.

关于平方补数问题与除数问题

On Problem On the Additive Square Complements and Its Divisor Problem

关于平方补数除数函数的均值

On the mean value of divisor function for square complements

本文研究了互素的无平方因子数的计数问题。

We have studied the counting of coprime square-free numbers in this paper .

简单随机样本平方平均数的分布

On the distribution of root - mean - square value of simple random sample

总体平方平均数的估计

On the estimation of population square mean

研究了平方补数集合中的除数问题，改进并推广了原有的结果。

The divisor problem in additive square complements is studied , which improves results ever deduced .

关于正整数的加法平方补数

On the additive square complements

平方补数的一个性质

A property of square complements

研究了平方剩余数的性质，给出了平方剩余数对于除数函数的均值公式。

Studying the properties of square residues , and giving a mean value formula of the divisor function for square residue sequences .

关于十进制非零数字之积函数的几何均值及数论函数对平方补数的均值

The Geometric Mean Value of Product of Non-zero Digits in Base 10 and the Mean Value of Some Number Functions for Square Complements

利用解析方法研究了平方剩余数的若干性质，并给出了平方剩余数对于两个常用数论函数的几个均值公式。

By using the analytic method some properties of the square residues are studied , and several ( interesting ) mean value formulas are given .

Hermite多项式平方及组合数的一组计算公式

Some Calculation Formulas for Square of Hermite Polynomials and Combination Categories

施磷量的增加有利于提高平方米穗数、穗粒数和结实率，但不利于千粒重的提高；

With the increase of fertilizer P , the number of spikes grains of per squre meter and cropping rate increased , but the weight of thousand grains decreased ;

随着播期的推迟每平方米穗数、穗粒数和千粒重均呈先上升后降低的趋势，两个品种的最高产量均出现在第二播期。

Fourthly , the spike number , kernel numbers per spike , and the weight of 1000 kernels increased firstly and then decreased with the sowing time postponed , and the maximum yields of the two varieties occurred at the second sowing time .

方法：对22例经病理证实为血管瘤的彩色多普勒超声表现进行回顾性分析。检测其血管密度（每平方厘米内血管数）、动脉收缩期峰值血流速度（PASV）、阻力指数（RI）等。

Method : The vessel density ( number of vessels per square centimeter ), peak arterial systolic flow velocity ( PASV ) and resistive index ( RI ) were examined by color Doppler ultrasonography in 22 hemangiomas proved by pathology .