整数的

Determine the greatest common divisor and least common multiple of a pair of integers.

请计算两个整数的最大公约数和最小公倍数。

GCD and LCM. Determine the greatest common divisor and least common multiple of a pair of integers.

最大公约数和最小公积数请计算两个整数的最大公约数和最小公倍数。

In this paper, we introduce the algorithm of Schoof-Elkies-Atkin to compute the order of elliptic curves over finite fields. We give out a fast algorithm to compute the division polynomial f〓 and a primitive point of order 2〓. This paper also gives an improved algorithm in computing elliptic curve scalar multiplication. Using the method of complex multiplication, we find good elliptic curves for use in cryptosystems, and implemented ElGamal public-key scheme based on elliptic curves. As a co-product, we also realized the algorithm to determine primes using Goldwasser-Kilian's theorem. Lastly, the elliptic curve method of integer factorization is discussed. By making some improvement and through properly selected parameters, we successfully factored an integer of 55 digits, which is the product of two 28-digit primes.

本文介绍了计算有限域上椭圆曲线群的阶的Schoof-Elkies-Atkin算法，在具体处理算法过程中，我们给出了计算除多项式f〓的快速算法和寻找2〓阶本原点的快速算法；标量乘法是有关椭圆曲线算法中的最基本运算，本文对[Koe96]中的椭圆曲线标量乘法作了改进，提高了其运算速度；椭圆曲线的参数的选择直接影向到椭圆曲线密码体的安全性，文中利用复乘方法构造了具有良好密码特性的椭圆曲线，并实现了椭圆曲线上ElGamal公钥体制；文中还给出了利用Goldwasser-Kilian定理和椭圆曲线的复乘方法进行素数的确定判别算法；最后讨论了利用椭圆曲线分解整数的方法并进行了某些改进，在PC机上分解了两个28位素数之积的55位整数。

If S is an infinite sequence of positive integers,is S an alpha sequence?

假如S是一个正整数的有限数列，那S是alpha数列吗？

The Integer Solutions Property: As long as all the supplies, demands, and arc capacities have integer values, any minimum-cost flow problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its flow quantities.

具有整数解的特征：只要其所有的供应、需求和弧的容量都是整数值，那么任何最小费用流问题的可行解就一定有所有流量都是整数的最优解

In chapter two, concepts of integrity, integral set and integral number are introduced firstly, and then some relationships between the integrity and some other parameters of graphs are given, then the upper and lower bounds of the integrity are investigated on the basis of integral number determined. The final result is the upper and lower bounds of integral number if the integrity is known.

第二部分主要讨论了完整度与图的其它参数之间的关系；给出了完整度给定条件下完整数的最大可能值及完整数给定条件下，完整度的最大最小可能值；给出了图的完整度与此图的删点子图的完整度之间的关系。

Consider for example a program that is specified as only having to work with values in the range 0 to 25000; a programmer might reasonably choose to use two-byte integers to represent such values. However, a calculation like "work out 85% of 24760" could cause problems, even though the result (21046) is in range. If the calculation is done by multiplying 85 and 24760, the intermediate result 2104600 is out of range.

考虑一个例子：一个程序被详说明其所处理的数值的范围是0—25000，一个编程者可能选择2-字节整数来表现这个值，然而，一个如，"算出24760的85%是多少"的计算可能产生问题，即使结果（21046）在范围之内，因为如果计算的过程中需要用85乘以24760，那么这个中间值——2104600超出了2-字节整数的范围。

Each transmitter will be assigned one positive integer channel.

每个传输器都会被分派一个正整数的信道。

The order of Fibonacci proper fraction: The row/column of the partition table is, even more amazing, exactly the order when sorting the Fibonacci proper fraction with fixed denominator/numerator.

二 正整数的分割。所有的移动步骤将正整数做了一个新的分割；此分割模k之后有良好的循环性质。费波那契真分数的排序。

In C under the TSA Sanchez Theorem : A sum of the cube can be written a bunch of odd that the sum...

在C下实现的尼科梅切斯定理：任何一个整数的立方都可以写成一串奇数之和即任意一个正整数n的立方分解成个连续的n个奇数之和。

I can not make it blossom and suits me

我不能让树为我开花

When temperatures are above approximately 80 °C discolouration of the raceways or rolling elements is a frequent feature.

当温度高于 80 °C 左右时，滚道或滚动元件褪色是很常见的特征。