In the history of mathematics different methods have been used to detect if a number is prime or not. In this paper a new one will be shown. It will be demonstrated that if the following equation is zero for a certain...In the history of mathematics different methods have been used to detect if a number is prime or not. In this paper a new one will be shown. It will be demonstrated that if the following equation is zero for a certain number p, this number p would be prime. And being m an integer number higher than (the lowest, the most efficient the operation). . If the result is an integer, this result will tell us how many permutations of two divisors, the input number has. As you can check, no recurrent division by odd or prime numbers is done, to check if the number is prime or has divisors. To get to this point, we will do the following. First, we will create a domain with all the composite numbers. This is easy, as you can just multiply one by one all the integers (greater or equal than 2) in that domain. So, you will get all the composite numbers (not getting any prime) in that domain. Then, we will use the Fourier transform to change from this original domain (called discrete time domain in this regards) to the frequency domain. There, we can check, using Parseval’s theorem, if a certain number is there or not. The use of Parseval’s theorem leads to the above integral. If the number p that we want to check is not in the domain, the result of the integral is zero and the number is a prime. If instead, the result is an integer, this integer will tell us how many permutations of two divisors the number p has. And, in consequence information how many factors, the number p has. So, for any number p lower than 2m?- 1, you can check if it is prime or not, just making the numerical definite integration. We will apply this integral in a computer program to check the efficiency of the operation. We will check, if no further developments are done, the numerical integration is inefficient computing-wise compared with brute-force checking. To be added, is the question regarding the level of accuracy needed (number of decimals and number of steps in the numerical integration) to have a reliable result for large numbers. This will be commented on the paper, but a separate study will be needed to have detailed conclusions. Of course, the best would be that in the future, an analytical result (or at least an approximation) for the summation or for the integration is achieved.展开更多
为满足地基大口径主焦点式望远镜的多谱段观测需求,对比研究了滤光片切换机构的结构形式。针对传统形式滤光轮机构的缺点,设计了一种沿光轴圆周均布式的新型滤光轮机构,以实现将不同透过谱段的滤光片依次在光路中切入和切出。该滤光轮...为满足地基大口径主焦点式望远镜的多谱段观测需求,对比研究了滤光片切换机构的结构形式。针对传统形式滤光轮机构的缺点,设计了一种沿光轴圆周均布式的新型滤光轮机构,以实现将不同透过谱段的滤光片依次在光路中切入和切出。该滤光轮机构由驱动组件和滤光片组件组成,与传统形式滤光轮机构相比,该机构安装时无需与光轴偏置,充分利用了光轴圆周空间,能够减小对主镜的遮拦。首先,介绍了滤光轮机构的组成及工作原理,并对关键部件进行了选型计算。随后,对滤光轮机构进行了模态分析,其一阶固有频率为108 Hz。最后,对滤光轮机构进行了切换精度测量、滤光片面形精度检测以及高低温试验。检测和试验结果表明,该滤光轮机构的滤光片切入后偏心误差最大为0.1 mm,滤光片面形精度均方根(Root Mean Square,RMS)值均优于λ/30,在系统的工作温度范围内切换顺畅无卡滞。该滤光轮机构能够满足大口径主焦点式望远镜的多谱段观测要求,为滤光轮设计提供了新思路。展开更多
文摘In the history of mathematics different methods have been used to detect if a number is prime or not. In this paper a new one will be shown. It will be demonstrated that if the following equation is zero for a certain number p, this number p would be prime. And being m an integer number higher than (the lowest, the most efficient the operation). . If the result is an integer, this result will tell us how many permutations of two divisors, the input number has. As you can check, no recurrent division by odd or prime numbers is done, to check if the number is prime or has divisors. To get to this point, we will do the following. First, we will create a domain with all the composite numbers. This is easy, as you can just multiply one by one all the integers (greater or equal than 2) in that domain. So, you will get all the composite numbers (not getting any prime) in that domain. Then, we will use the Fourier transform to change from this original domain (called discrete time domain in this regards) to the frequency domain. There, we can check, using Parseval’s theorem, if a certain number is there or not. The use of Parseval’s theorem leads to the above integral. If the number p that we want to check is not in the domain, the result of the integral is zero and the number is a prime. If instead, the result is an integer, this integer will tell us how many permutations of two divisors the number p has. And, in consequence information how many factors, the number p has. So, for any number p lower than 2m?- 1, you can check if it is prime or not, just making the numerical definite integration. We will apply this integral in a computer program to check the efficiency of the operation. We will check, if no further developments are done, the numerical integration is inefficient computing-wise compared with brute-force checking. To be added, is the question regarding the level of accuracy needed (number of decimals and number of steps in the numerical integration) to have a reliable result for large numbers. This will be commented on the paper, but a separate study will be needed to have detailed conclusions. Of course, the best would be that in the future, an analytical result (or at least an approximation) for the summation or for the integration is achieved.
文摘为满足地基大口径主焦点式望远镜的多谱段观测需求,对比研究了滤光片切换机构的结构形式。针对传统形式滤光轮机构的缺点,设计了一种沿光轴圆周均布式的新型滤光轮机构,以实现将不同透过谱段的滤光片依次在光路中切入和切出。该滤光轮机构由驱动组件和滤光片组件组成,与传统形式滤光轮机构相比,该机构安装时无需与光轴偏置,充分利用了光轴圆周空间,能够减小对主镜的遮拦。首先,介绍了滤光轮机构的组成及工作原理,并对关键部件进行了选型计算。随后,对滤光轮机构进行了模态分析,其一阶固有频率为108 Hz。最后,对滤光轮机构进行了切换精度测量、滤光片面形精度检测以及高低温试验。检测和试验结果表明,该滤光轮机构的滤光片切入后偏心误差最大为0.1 mm,滤光片面形精度均方根(Root Mean Square,RMS)值均优于λ/30,在系统的工作温度范围内切换顺畅无卡滞。该滤光轮机构能够满足大口径主焦点式望远镜的多谱段观测要求,为滤光轮设计提供了新思路。
