Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers

A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: MN = 2N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;MN]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

Effect of Richardson number on entropy generation over backward facing step

The flow over a backward facing step （BFS） has been taken as a useful proto- type to investigate intrinsic mechanisms of separated flow with heat transfer. However, to date, the open literature on the effect of Richardson number on entropy generation over the BFS is absent yet, although the flow pattern and heat transfer characteristic both will receive significant influence caused by the variation of Richardson number in many prac- tical applications, such as in microelectromechanical systems and aerocrafts. The effect of Richardson number on entropy generation in the BFS flow is reported in this paper for the first time. The entropy generation analysis is conducted through numerically solving the entropy generation equation. The velocity and temperature, which are the inputs of the entropy generation equation, are evaluated by the lattice Boltzmann method. It is found that the distributions of local entropy generation number and Bejan number are significantly influenced by the variation of Richardson number. The total entropy gen- eration number is a monotonic decreasing function of Richardson number, whereas the average Bejan number is a monotonic increasing function of Richardson number.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

Improved quantum randomness amplification with finite number of untrusted devices based on a novel extractor

Quantum randomness amplification protocols have increasingly attracted attention tbr their tantastic ability to ampllI~, weak randomness to almost ideal randomness by utilizing quantum systems. Recently, a realistic noise-tolerant randomness amplification protocol using a finite number of untrusted devices was proposed. The protocol has the composable security against non-signalling eavesdroppers and could produce a single bit of randomness from weak randomness sources, which is certified by the violation of certain Bell inequalities. However, the protocol has a non-ignorable limitation on the min- entropy of independent sources. In this paper, we further develop the randomness amplification method and present a novel quantum randomness amplification protocol based on an explicit non-malleable two independent-source randomness extractor, which could remarkably reduce the above-mentioned specific limitation. Moreover, the composable security of our improved protocol is also proposed. Our results could significantly expand the application range for practical quantum randomness amplification, and provide a new insight on the practical design method for randomness extraction.

Infinite Number of Twin Primes

This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.

400 Gb/s physical random number generation based on deformed square self-chaotic lasers

A circular-sided square microcavity laser etched a central hole has achieved chaos operation with a bandwidth of 20.8 GHz without external optical feedback or injection,in which the intensity probability distribution of a chaotic signal with a twopeak pattern was observed.Based on the self-chaotic microlaser,physical random numbers at 400 Gb/s were generated by extracting the four least significant bits without other complex post-processing methods.The solitary chaos laser and minimal post-processing have predicted a simpler and low-cost on-chip random number generator in the future.

Random-injection-based two-channel chaos with enhanced bandwidth and suppressed time-delay signature by mutually coupled lasers: Proposal and numerical analysis

Simultaneous bandwidth(BW) enhancement and time-delay signature(TDS) suppression of chaotic lasing over a wide range of parameters by mutually coupled semiconductor lasers(MCSLs) with random optical injection are proposed and numerically investigated. The influences of system parameters on TDS suppression(characterized by autocorrelation function(ACF) and permutation entropy(PE) around characteristic time) and chaos BW are investigated. The results show that, with the increasing bias current, the ranges of parameters(detuning and injection strength) for the larger BW(> 20 GHz) are broadened considerably, while the parameter range for optimized TDS(< 0.1) is not shrunk obviously.Under optimized parameters, the system can simultaneously achieve two chaos outputs with enhanced BW(> 20 GHz)and perfect TDS suppression. In addition, the system can generate two-channel high-speed truly physical random number sequences at 200 Gbits/s for each channel.

Entropy analysis in electrical magnetohydrodynamic(MHD) flow of nanofluid with effects of thermal radiation,viscous dissipation,and chemical reaction

The unsteady mixed convection flow of electrical conducting nanofluid and heat transfer due to a permeable linear stretching sheet with the combined effects of an electric field, magnetic field, thermal radiation, viscous dissipation, and chemical reaction have been investigated. A similarity transformation is used to transform the constitutive equations into a system of nonlinear ordinary differential equations.The resultant system of equations is then solved numerically using implicit finite difference method.The velocity, temperature, concentration, entropy generation, and Bejan number are obtained with the dependence of different emerging parameters examined. It is noticed that the velocity is more sensible with high values of electric field and diminished with a magnetic field. The radiative heat transfer and viscous dissipation enhance the heat conduction in the system. Moreover, the impact of mixed convection parameter and Buoyancy ratio parameter on Bejan number profile has reverse effects. A chemical reaction reduced the nanoparticle concentration for higher values.

Development of a methodology for assessing the adequacy of electric power systems

This study presents the results of a research into the developing a methodology for assessing the adequacy of advanced electric power systems characterized by the integration of various innovative technologies,which complicates their analysis.The methodology development is aimed at solving two main problems:(1)increase the adequacy of modeling the processes that occur in the electric power system and (2)enhance the computational efficiency of the adequacy assessment methodology.This study proposes a new mathematical model to minimize the power shortage and enhance the adequacy of modeling the processes.The model considers quadratic power transmission losses and network coefficients.The computational efficiency of the adequacy assessment methodology is enhanced using efficient random-number generators to form the calculated states of electric power systems and machine learning methods to assess power shortages and other reliability characteristics in the calculated states.

Is Independent Choice Possible?

This paper questions the generally accepted assumption that one can make a random choice that is independent of the rest of the universe. We give a general description of any setup that could be conceived to generate random numbers. Based on the fact that the initial state of such setup together with its environment cannot be known, we show that the independence of its generated output cannot be guaranteed. Some consequences of this theoretical limitation are discussed.

Ultrafast and real-time physical random bit extraction with all-optical quantization

Optical chaos generated by perturbing semiconductor lasers has been viewed,over recent decades,as an excellent entropy source for fast physical random bit generation(RBG)owing to its high bandwidth and large random fluctuations.However,most optical-chaos-based random bit generators perform their quantization process in the electrical domain using electrical analog-to-digital converters,so their real-time rates in a single channel are severely limited at the level of Gb/s due to the electronic bottleneck.Here,we propose and experimentally demonstrate an all-optical method for RBG where chaotic pulses are quantized into a physical random bit stream in the all-optical domain by means of a length of highly nonlinear fiber.In our proof-of-concept experiment,a 10-Gb/s random bit stream is successfully generated on-line using our method.Note that the single-channel real-time rate is limited only by the chaos bandwidth.Considering that the Kerr nonlinearity of silica fiber with an ultrafast response of few femtoseconds is exploited for composing the key part of quantizing laser chaos,this scheme thus may operate potentially at much higher real-time rates than 100 Gb/s provided that a chaotic entropy source of sufficient bandwidth is available.

Loss of time-delay signature in a ring of three unidirectionally coupled semiconductor lasers

A ring of three unidirectionally coupled semiconductor lasers （RTUC-SLs） is used to generate broadband chaos with no pronounced time-delay （TD） signature. Using the autocorrelation function and pernmtation entropy as the TD measures, we demonstrate that under suitable coupling strength, the loss of the TD signature of the lasers in the RTUC-SL configuration is achieved both for the intensity and the phase. These findings should prove valuable for developing high-quality optical chaos for potential applications, such as chaos-based communica- tions and random number generation.