Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number

We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.

Preliminary Identification of a Prime Number Other Than 2 and 3, the Origin of Twin Prime Numbers, the Structure of the Chain of Prime Numbers and the Set of Prime Numbers Less Than a Given Integer

The application of the Euclidean division theorem for the positive integers allowed us to establish a set which contains all the prime numbers and this set we called it set of supposedly prime numbers and we noted it Esp. We subsequently established from the previous set the set of non-prime numbers (the set of numbers belonging to this set and which are not prime) denoted Enp. We then extracted from the set of supposedly prime numbers the numbers which are not prime and the set of remaining number constitutes the set of prime numbers denoted Ep. We have deduced from the previous set, the set of prime numbers between two natural numbers. We have explained during our demonstrations the origin of the twin prime numbers and the structure of the chain of prime numbers.

How Prime Numbers Are Interconnected and Built with Two Equations: Addition and Subtraction Rules the Function (6µ)

Are all prime numbers linked by four simple functions? Can we predict when a prime will appear in a sequence of primes? If we classify primes into two groups, Group 1 for all primes that appear before ζ (such that , for instance 5, ), an even number divisible by 3 and 2, and Group 2 for all primes that are after ζ (such that , for instance 7), then we find a simple function: for each prime in each group, , where n is any natural number. If we start a sequence of primes with 5 for Group 1 and 7 for Group 2, we can attribute a μ value for each prime. The μ value can be attributed to every prime greater than 7. Thus for Group 1, and . Using this formula, all the primes appear for , where μ is any natural number.

Lychrel Numbers in Base 10: A Probabilistic Approach

For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.

On Perron’s Formula and the Prime Numbers

The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the Riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form limT∈Ν→∞fT(zw)=zw,where, zwis a real number.

Three- and Four-Dimensional Generalized Pythagorean Numbers

The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a2+b2=c2 with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a2+b2+c2=d2 with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d2 is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a2+b2+c2+d2=e2 and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d2, respectively, e2.

Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy,accessibility,and cost-effectiveness.This paper investigates the potential applications of intuitionistic fuzzy sets(IFS)with rough sets in the context of sparse data.When it comes to capture uncertain information emanating fromboth upper and lower approximations,these intuitionistic fuzzy rough numbers(IFRNs)are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets,respectively.We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties.We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles.These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator(IFRWFA)and intuitionistic fuzzy rough ordered weighted fairly aggregation operator(IFRFOWA).These operators successfully adjust to membership and non-membership categories with fairness and subtlety.We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multiattribute decision-making field.We use the intuitionistic fuzzy rough environment’s architecture to create a novel strategy in situation involving several decision-makers and non-weighted data.Additionally,we developed a novel technique by combining the IFSs with quaternion numbers.We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers(IFQNs).With the help of this framework,we can simulate uncertainty in real-world situations and address a number of decision-making problems.Using the examples we have released,we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity ofmedical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology.

Effect of Sweptback Angle of a Delta Wing on Surface Pressure Distribution at Supersonic Mach Numbers

The purpose of this work is to shed light on the effect of the pivot position on the surface pressure distribution over a 3D wing in different flight conditions.The study is intended to support the design and development of aerospace vehicles where stability analysis,performance optimization,and aircraft design are of primary importance.The following parameters are considered:Mach numbers(M)of 1.3,1.8,2.3,2.8,3.3,and 3.8,angle of incidence(θ)in the range from 5°to 25°,pivot position from h=0.2 to 1.The results of the CFD numerical simulations match available analytical data,thereby providing evidence for the reliability of the used approach.The findings provide valuable insights into the relationship between the surface pressure distribution,the Mach number and the angle of incidence.

Combinatorial Identities Concerning Harmonic Numbers

In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.

Distribution and Relation of Primes with Tesla Numbers 3, 6, and 9

This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Conjecture problem. We expose a peculiar relation between twin primes and the generation of prime numbers with Tesla numbers. Tesla numbers seem to be present in so many domains like time, vibration and frequency [1], and the space between twin primes is not the exception. Let us say that twin primes are more than just prime numbers plus 2 or minus 2, and Tesla numbers are more involved with twin primes than we think, and hopefully, this approach give us a better understanding of the distribution of the twin pairs.

Natural Numbers and the Strong Goldbach Conjecture

This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.

Flow and sound fields of scaled high-speed trains with different coach numbers running in long tunnel

Segregated incompressible large eddy simulation and acoustic perturbation equations were used to obtain the flow field and sound field of 1:25 scale trains with three,six and eight coaches in a long tunnel,and the aerodynamic results were verified by wind tunnel test with the same scale two-coach train model.Time-averaged drag coefficients of the head coach of three trains are similar,but at the tail coach of the multi-group trains it is much larger than that of the three-coach train.The eight-coach train presents the largest increment from the head coach to the tail coach in the standard deviation(STD)of aerodynamic force coefficients:0.0110 for drag coefficient(Cd),0.0198 for lift coefficient(Cl)and 0.0371 for side coef-ficient(Cs).Total sound pressure level at the bottom of multi-group trains presents a significant streamwise increase,which is different from the three-coach train.Tunnel walls affect the acoustic distribution at the bottom,only after the coach number reaches a certain value,and the streamwise increase in the sound pressure fluctuation of multi-group trains is strengthened by coach number.Fourier transform of the turbulent and sound pressures presents that coach number has little influence on the peak frequencies,but increases the sound pressure level values at the tail bogie cavities.Furthermore,different from the turbulent pressure,the first two sound pressure proper orthogonal decomposition(POD)modes in the bogie cavities contain 90%of the total energy,and the spatial distributions indicate that the acoustic distributions in the head and tail bogies are not related to coach number.

Identities on q-Harmonic Numbers

With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.

Z-numbers的否定及其在多属性决策中的应用

否定命题是数理逻辑中的重要表示方式,也是智能信息处理中刻画信息的关键工具.Znumber是一种新的不确定性语言信息表示方式.回顾了否定的相关研究,并介绍了可能性分布的否定在Z-number理论中的应用.通过最大香农熵和粒子群算法,提出了一种优化模型,用于发现Z-number中隐藏的可能性分布.研究表明,Z-number的否定为语言决策分析和解决复杂问题提供了有效工具.

Cardinal Numbers of Fuzzy Sets

Up to now, the study on the cardinal number of fuzzy sets has advanced at on pace since it is very hard to give it an appropriate definition. Althrough for it in [1], it is with some harsh terms and is not reasonable as we point out in this paper. In the paper, we give a general definition of fuzzy cardinal numbers. Based on this definition, we not only obtain a large part of results with re spect to cardinal numbers, but also give a few of new properties of fuzzy cardinal numbers.

FLOW FIELD ANALYSES OF PLANE JET AT LOW REYNOLDS NUMBERS USING LATTICE BOLTZMANN METHOD

A two-dimensional（2-D） incompressible plane jet is investigated using the lattice Boltzmann method（LBM） for low Reynolds numbers of 42 and 65 based on the jet-exit-width and the maximum jet-exit-velocity. The results show that the mean centerline velocity decays as x-1/3 and the jet spreads as x2/3 in the self-similar region, which are consistent with the theoretical predictions and the experimental data. The time histories and PSD analyses of the instantaneous centerline velocities indicate the periodic behavior and the interaction between periodic components of velocities should not be neglected in the far field region, although it is invisible in the near field region.

NOVEL GENERALIZED GREY INCIDENCE MODEL BASED ON INTERVAL GREY NUMBERS

To extend the traditional generalized grey incidence model, a novel grey incidence model based on inter- val grey numbers is constructed. Considering the numerical information of indexes cannot be accurately obtained and can be defined as interval grey numbers, the interval grey numbers are defined as standard interval grey num- bers which are split in white part and grey part. The absolute degree of incidence and relative degree of incidence based on the interval grey numbers are constructed and their arithmetic are given. Finally, an example about commercial aircraft index selection illuminates the effectiveness of the model. The results show that the model can sort indexes better and can extend the grey incidence models significantly.

Ramsey numbers r(K_(1, 4), G) for all three-partite graphs G of order six

In this paper, we use a combinatorial analysis method. In the complete graph K N with edges colored arbitrarily by red or blue, we consider the proposition of the subgraph of the red graph or blue graph induced by the neighborhood of some vertex in V(K N). Inspired by the main results of Jayawardene and Rousseau (Ars Combinatoria, 2000, 163-173), we determine the Ramsey numbers of r(K 1, 4, G), where G is the three-partite graph of order six without isolate vertex.

Asymptotic upper bounds for wheel:complete graph Ramsey numbers

It is shown that r(W_m, K_n)≤(1+o(1))C_1n log n 2m-2m-2 for fixed even m≥4 and n→∞, and r(W_m, K_n)≤(1+o(1))C_2n 2mm+1 log n m+1m-1 for fixed odd m≥5 and n→∞, where C_1=C_1(m)>0 and C_2=C_2(m)>0, in particular, C_2=12 if m=5 . It is obtained by the analytic method and using the function f_m(x)=∫ 1 _ 0 (1-t) 1m dtm+(x-m)t , x≥0 , m≥1 on the base of the asymptotic upper bounds for r(C_m, K_n) which were given by Caro, et al. Also, cn log n 52 ≤r(K_4, K_n)≤(1+o(1)) n 3 ( log n) 2 (as n→∞ ). Moreover, we give r(K_k+C_m, K_n)≤(1+o(1))C_5(m)n log n k+mm-2 for fixed even m≥4 and r(K_k+C_m, K_n)≤(1+o(1))C_6(m)n 2+(k+1)(m-1)2+k(m-1) log n k+2m-1 for fixed odd m≥3 (as n→∞ ).

新型犹豫不确定Z-Numbers熵及其多准则群决策分析

犹豫不确定语言型Z-Numbers的语言尺度函数是由两种不同语义的集合构成,分别度量两种语言尺度函数,确定其三个基本性质。结合语言尺度函数和犹豫不确定语言型Z-Numbers的模糊因子及犹豫因子,构造了犹豫不确定语言型Z-Numbers的交叉熵和熵,引入求解决策者权重和准则权重的两种最优化模型,通过求解最优化模型确定综合熵决策矩阵,由排序值公式给出多准则群决策问题的最优方案。通过一个实例分析比较得出相同的最优方案和不同排序,验证了该决策方法的有效性与可行性。