费马小定理的几何意义

费马小定理是数论中的一个重要定理,但对其几何意义的论述几乎没有,给出了费马小定理的几何意义。     

A Note on Strong Dickson Pseudoprimes

In 1994 the infinitude of Fermat pseudoprimes to any bases (i.e. Carmichael numbers) has been established (cf. [2, 42]), the smallest being 561 = 3 ċ 11 ċ 17. If, instead of the power function the Dickson polynomial, respectively Lucas sequence, V _n (P, Q) ≡P (mod n), is being used as the primality testing function for P, Q∈ℤ, then the smallest pseudoprime to this test with respect to all parameters P and Q is 443372888629441 = 17 ċ 31 ċ 41 ċ 43 ċ 89 ċ 97 ċ 167 ċ 331. Not more than about 50 of such strong Dickson pseudoprimes are known at present, each of them having a more complex structure than the (Fermat-) Carmichael numbers. Those particular properties will be the focus of our attention in this note. After summarizing some results on strong Dickson pseudoprimes their connections to other types of pseudoprimes are described and characterizations of superstrong Dickson pseudoprimes are presented. Furthermore, an algorithm for an effective generation of those kinds of numbers is introduced. Received: April 25, 1997; revised version: July 20, 1998     

求模逆元的几种算法

基于模乘法逆元的定义、存在条件及其相关定理,首先,对各求模逆元的算法思想和计算过程进行了深入的剖析,并总结了它们各自的运算特点以及它们的局限性所在,最后,依据可计算的复杂性理论和实际所测试的数据,比较了各种算法的执行效率以及它们的使用范围。     

Fermat方程有正整数解的两个必要条件

本文给出了不定方程x~p+y~p=z~p,pxyz有正整数解的两个必要条件,其中p为任意奇素数.     

对射线传输系统的推广及应用

基于高频射线理论,并通过Fermat原理以一致的形式讨论了左、右手媒质界面上电波或光波的折射特性．在所得折射定律的基础上结合引入相位的辛空间坐标,对矩阵光学进行了推广．相应传输理论的表示仍然保持了Gauss理论的辛几何本质,从而可以适用于左、右手媒质混合的系统．     

Fermat原理求解柱-锥系统二阶反射问题

以圆柱和圆锥二阶系统的反射寻迹为例,结合费马原理和反射定律,推导出各反射点Z坐标的解析表达式,从而减少了寻迹方程中未知数个数,使计算时间显著缩短,也提高了结果精度,文中给出了应用实例．利用类似的方法可以解决其他类型的二阶反射寻迹问题,也对高阶反射问题求解起一定的参考作用．     

Building pseudoprimes with a large number of prime factors

We extend the method due originally to Löh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.On leave from the French Department of Defense, Délégation Générale pour l'Armement     

On the existence of solutions to a class of minimum time control problems and applications to Fermat's Principle and to the Brachystocrone

We prove a theorem for the existence of solutions to minimum time control problems, under assumptions that do not require the convexity of the images and that weaken the assumption of upper semicontinuity. Our result applies to Fermat's Principle and to the Brachystocrone.     

On strong Dickson pseudoprimes

It is known that the Lucas sequenceV _n(,c)=aⁿ + bⁿ,a, b being the roots ofx ² – x + c=0 equals the Dickson polynomial .^n–2i Lidl, Müller and Oswald recently defined a number b to be a strong Dickson pseudoprime to the parameterc (shortlysDpp(c)) if [itg_n(b, c)b modn for all b. These numbers seem to be very appropriate for a fast probabilistic prime number test. In generalizing results of the above mentioned authors a criterion is derived for an odd composite number to be ansDpp(c) for fixedc. Furthermore the optimal parameterc for the prime number test is determined.     

最小时间两点射线追踪

本文利用费马原理提出了一种快速、有效的射线追踪方法－－最小时间射线追踪方法。首先利用三次样条函数详细描述地质模型，根据费马原理导出微分形式的射线路径方程组。为确保算法的鲁棒性和精确度，本文提出在保证一定运算速度的情况下，尽可能同时提供多种解法，以确保成功地得到射线路径，并且给出多组不同的初始值，以解决多解问题；然后以利用梯度法为例简单介绍了求解的过程，了梯度法的主要数值计算公式。模型试算结果表明，本文所提出的方法具有精确、适用于复杂地质模型的特点，其运算速度与打靶法相比提高了两个数量级。