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.

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.

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.

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.

The prediction of donor number and acceptor number of electrolyte solvent molecules based on machine learning

Electrolyte solvents have a critical impact on the design of high performance and safe batteries.Gutmann's donor number(DN) and acceptor number(AN) values are two important parameters to screen and design superior electrolyte solvents. However, it is more time-consuming and expensive to obtain DN and AN values through experimental measurements. Therefore, it is essential to develop a method to predict DN and AN values. This paper presented the prediction models for DN and AN based on molecular structure descriptors of solvents, using four machine learning algorithms such as Cat Boost(Categorical Boosting), GBRT(Gradient Boosting Regression Tree), RF(Random Forest) and RR(Ridge Regression).The results showed that the DN and AN prediction models based on Cat Boost algorithm possesses satisfactory prediction ability, with R^(2) values of the testing set are 0.860 and 0.96. Moreover, the study analyzed the molecular structure parameters that impact DN and AN. The results indicated that TDB02m(3D Topological distance based descriptors-lag 2 weighted by mass) had a significant effect on DN, while HATS1s(leverage-weighted autocorrelation of lag 1/weighted by I-state) plays an important role in AN. The work provided an efficient approach for accurately predicting DN and AN values, which is useful for screening and designing electrolyte solvents.

A novel Effective Panicle Number per Plant 4 haplotype enhances grain yield by coordinating panicle number and grain number in rice

Increasing effective panicle number per plant(EPN)is one approach to increase yield potential in rice.However,molecular mechanisms underlying EPN remain unclear.In this study,we integrated mapbased cloning and genome-wide association analysis to identify the EPN4 gene,which is allelic to NARROW LEAF1(NAL1).Overexpression lines containing the Teqing allele(TQ)of EPN4 had significantly increased EPN.NIL-EPN4^(TQ) in japonica(geng)cultivar Lemont(LT)exhibited significantly improved EPN but decreased grain number and flag leaf size relative to LT.Haplotype analysis indicated that accessions with EPN4-1 had medium EPN,medium grain number,and medium grain weight,but had the highest grain yield among seven haplotypes,indicating that EPN4-1 is an elite haplotype of EPN4 for positive coordination of the three components of grain yield.Furthermore,accessions carrying the combination of EPN4-1 and haplotype GNP1-6 of GNP1 for grain number per panicle showed higher grain yield than those with other allele combinations.Therefore,pyramiding of EPN4-1 and GNP1-6 could be a preferred approach to obtain high yield potential in breeding.

The wheat sucrose synthase gene TaSus1 is a determinant of grain number per spike

Some haplotypes of the sucrose synthase gene TaSus1 are associated with thousand-grain weight(TGW)in wheat(Triticum aestivum L.).However,no mutations have been identified within the gene to test this association.The effects of TaSus1 on grain number per spike(GNS)also are largely unknown.Our previous genome-wide association study identified TaSus-A1 as a candidate gene controlling fertile spikelet number per spike(FSN).In the present study,we generated two independent mutants for the three TaSus1 homoeologs by CRISPR/Cas9-mediated genome editing.The triple mutants displayed lower FSN,GNS,grain number per spikelet(GNST),and TGW than wild-type plants.In 306 hexaploid wheat accessions,two single-nucleotide polymorphisms in TaSus-A1 contributed differently to GNS.Introgression of the two alleles into a wheat genetic background confirmed their effects.The alleles differed in geographical distribution among the accessions.

On the Number of Primes in the Interval (x, 2x) by an Elementary Method

An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite numbers distributed in the interval [1, 2x]). An elementary method to know the number of primes in a given magnitude is suitably placed in the form of a general formula, and we have proved it. The general formula is applied to the terms of the equation, and a tactical simplification of the terms gives rise to an expression whose verification envisages scope for its further studies.

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.

A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .

A Number Theoretic Analysis of the Enthalpy, Enthalpy Energy Density, Thermodynamic Volume, and the Equation of State of a Modified White Hole, and the Implications to the Quantum Vacuum Spacetime, Matter Creation and the Planck Frequency

In this paper, we analyze the enthalpy, enthalpy energy density, thermodynamic volume, and the equation of state of a modified white hole. We obtain new possible mathematical connections with some sectors of Number Theory, Ramanujan Recurring Numbers, DN Constant and String Theory, that enable us to extract the quantum geometrical properties of these thermodynamic equations and the implication to the quantum vacuum spacetime geometry of our early universe as they act as the constraints to the nature of quantum gravity of the universe.

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.

On the Expression of Composite Natural Numbers as Tensorial Products of Prime Natural Powers Vectors

This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.

基于决策者偏好构造的概率语言型Z-number多属性决策方法

由于决策环境的日益复杂,专家评价信息往往会包含专家偏好,因此有必要对专家偏好进行识别与处理。针对这一情况,提出了基于决策者偏好构造的概率语言型Z-number并应用于多属性决策问题中。首先,针对专家偏好未知的情况,提出了一种基于修正的平均得分距离,由专家评价信息确定专家偏好。然后,依据得到的专家偏好,利用GOWPA算子与最大熵生成概率语言型Z-number。同时,获得专家评价信息的可信度,由专家可信度聚合专家信息并采用TODIM方法对方案进行排序。最后,在国家大力提倡节能减排的背景下,将所提方法应用于国家绿色矿山试点企业的选取上,验证了所提方法的有效性和可行性。

Genome-wide investigation to assess copy number variants in the Italian local chicken population

Background Copy number variants(CNV)hold significant functional and evolutionary importance.Numerous ongoing CNV studies aim to elucidate the etiology of human diseases and gain insights into the population structure of livestock.High-density chips have enabled the detection of CNV with increased resolution,leading to the identification of even small CNV.This study aimed to identify CNV in local Italian chicken breeds and investigate their distribution across the genome.Results Copy number variants were mainly distributed across the first six chromosomes and primarily associated with loss type CNV.The majority of CNV in the investigated breeds were of types 0 and 1,and the minimum length of CNV was significantly larger than that reported in previous studies.Interestingly,a high proportion of the length of chromosome 16 was covered by copy number variation regions(CNVR),with the major histocompatibility complex being the likely cause.Among the genes identified within CNVR,only those present in at least five animals across breeds(n=95)were discussed to reduce the focus on redundant CNV.Some of these genes have been associated to functional traits in chickens.Notably,several CNVR on different chromosomes harbor genes related to muscle development,tissue-specific biological processes,heat stress resistance,and immune response.Quantitative trait loci(QTL)were also analyzed to investigate potential overlapping with the identified CNVR:54 out of the 95 gene-containing regions overlapped with 428 QTL associated to body weight and size,carcass characteristics,egg production,egg components,fat deposition,and feed intake.Conclusions The genomic phenomena reported in this study that can cause changes in the distribution of CNV within the genome over time and the comparison of these differences in CNVR of the local chicken breeds could help in preserving these genetic resources.

The first factor affecting dryland winter wheat grain yield under various mulching measures: Spike number

Water is the key factor limiting dryland wheat grain yield.Mulching affects crop yield and yield components by affecting soil moisture.Further research is needed to determine the relationships between yield components and soil moisture with yield,and to identify the most important factor affecting grain yield under various mulching measures.A long-term 9-yearifeld experiment in the Loess Plateau of Northwest China was carried out with three treatments:no mulch (CK),plastic mulch (M_(P)) and straw mulch (M_(S)).Yield factors and soil moisture were measured,and the relationships between them were explored by correlation analysis,structural equation modeling and significance analysis.The results showed that compared with CK,the average grain yields of M_(P) and M_(S) increased by 13.0and 10.6%,respectively.The average annual grain yield of the M_(P) treatment was 134 kg ha^(–1) higher than the M_(S) treatment.There were no significant differences in yield components among the three treatments (P<0.05).Soil water storage of the M_(S) treatment was greater than the M_(P) treatment,although the differences were not statistically signifiant.Soil water storage during the summer fallow period (SWSSF) and soil water storage before sowing (SWSS) of M_(S) were significantly higher than in CK,which increased by 38.5 and 13.6%,respectively.The relationship between M_(P) and CK was not statistically significant for SWSSF,but the SWSS in M_(P) was significantly higher than in CK.In terms of soil water storage after harvest (SWSH) and water consumption in the growth period(ET),there were no signi?cant differences among the three treatments.Based on the three analysis methods,we found that spike number and ET were positively correlated with grain yield.However,the relative importance of spike number to yield was the greatest in the M_(P )and M_(S) treatments,while that of ET was the greatest in CK.Suifcient SWSSF could indirectly increase spike number and ET in the three treatments.Based on these results,mulch can improve yield and soil water storage.The most important factor affecting the grain yield of dryland wheat was spike number under mulching,and ET with CK.These findings may help us to understand the main factors influencing dryland wheat grain yield under mulching conditions compared to CK.

Generalizations of Euler Numbers and Euler Numbers of Higher Order

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.

Structured condition numbers and statistical condition estimation for the LDU factorization

In this article, we consider the structured condition numbers for LDU, factorization by using the modified matrix-vector approach and the differential calculus, which can be represented by sets of parameters. By setting the specific norms and weight parameters, we present the expressions of the structured normwise, mixed, componentwise condition numbers and the corresponding results for unstructured ones. In addition, we investigate the statistical estimation of condition numbers of LDU factorization using the probabilistic spectral norm estimator and the small-sample statistical condition estimation method, and devise three algorithms. Finally, we compare the structured condition numbers with the corresponding unstructured ones in numerical experiments.

基于Z-number-IBWM-TODIM的应急供应商评估与选择研究

文章针对应急供应商选择过程中评价信息的不完全可靠性及专家背景的多样性问题,提出了一种基于Z-number和IBWM-TODIM方法相结合的评估框架,以此来选择应急供应商。该方法利用Z-number同时描述认知信息的不确定性和可靠性,增强了决策结果的可靠性和合理性;使用解释结构模型ISM确定最优指标和最劣指标,降低BWM方法的主观性,得到了指标权重;通过群体一致性和Z-number中的可靠性信息确定专家权重,使用TODIM方法进行排序,为应急供应商选择提供了更为准确和可靠的决策依据。