Let a,b,c be positive integers such that a²+b²=c², 2|b,gcd(a,b)=1. In 1956, Jesmanowicz conjectured that for any positive integer w, the only solution of (aw)^x+(bw)^y=(cw)^z in positive integers is (x, y, z) = (2, 2, 2). In this paper, based on Gaussian integer ring, we show that Je$\acute{s}$manowicz' conjecture is true for any positive integer

In this research first, a sequence of properties called delta is assigned to each prime number and then examined. Deltas are only dependent on the distribution of prime numbers, so the results obtained for the delta distribution can be considered as a proxy for the distribution of prime numbers. The first observation was that these properties are not unique and different prime numbers may have the same value of delta of a given order. It was found that a small number of deltas cover a large portion of prime numbers, so by recognizing repetitive deltas, the next prime numbers can be predicted with a certain probability, but the most important observation of this study is the normal distribution of deltas. This research has not tried to justify the obtained observations and instead of answering the questions, it seeks to ask the right question.

A prime gap is the difference between consecutive prime numbers. The $n^{\text{th}}$ prime gap, denoted $g_{n}$, is calculated by subtracting the $n^{\text{th}}$ prime from the $(n+1)^{\text{th}}$ prime: $g_{n}=p_{n+1}-p_{n}$. Oppermann's conjecture is a prominent unsolved problem in pure mathematics concerning prime gaps. Despite verification for numerous primes, a general proof remains elusive. If true, the conjecture implies that prime gaps grow at a rate bounded by $g_{n}<{\sqrt {p_{n}}}$. This note presents a proof of Oppermann's conjecture using the square root function on inequalities involving prime numbers. This proof simultaneously establishes Andrica's, Legendre's, and Brocard's conjectures.

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

We develop a primality test for Mersenne numbers and investigate the primality of the sequence of Catalan-Mersenne numbers. We use induction as a proof to show that the 5th Catalan-Mersenne number is a prime number, thus demonstrating the infinitude of Mersenne primes.

We developed some questions regarding twin prime conjecture by examining the bounded gaps between composite numbers and explained the relations between the formulas regarding composite numbers. Then we proved the twin prime conjecture to answer the new questions raised by proving the expression: \[ \liminf_{n\to\infty} (c_n_+_1 - c_n) \ge 10. \] In addition, we prove many different approaches have been and it has been proved that there must be a twin prime in the range $(p_k_2\cdot p_k_1, p^2_k_1)$, where $p \neq 2$.

In this paper, we study the symmetry between the supersingular prime number according to smallest sphenic number. With this symmetry, we show that the elements of sporadic group generates all prime numbers with the order by a simple application.

We talk about random when it is not possible to determine a pattern on the observed outcomes. A computer follows a sequence of fixed instructions to give any of its output, hence the difficulty of choosing numbers randomly from algorithmic approaches. However, some algorithms based on mathematical formulas like the Linear Congruential algorithm and the Lagged Fibonacci generator appear to produce "true" random sequences to anyone who does not know the secret initial input [1]. Up to now, we cannot rigorously answer the question on the randomness of prime numbers [2, page 1] and this highlights a connection between random number generator and the distribution of primes. From [3] and [4] one sees that it is quite naive to expect good random reproduction with prime numbers. We are, however, interested in the properties underlying the distribution of prime numbers, which emerge as sufficient or insufficient arguments to conclude a proof by contradiction which tends to show that prime numbers are not randomly distributed. To achieve this end, we use prime gap sequence variation. The algorithm that we produce makes possible to deduce, in the case of a binary choice, a uniform behavior in the individual consecutive occurrence of primes, and no uniformity trait when the occurrences are taken collectively.

I had previously developed an approach called Gocgen approach, which claimed to prove twin prime conjecture. In this paper, I processed the previously developed Gocgen approach with the zeta function and explained the relationship between the zeta function with some formulas to offer another perspective on the zeta function, and also created a new function that explains the Gocgen approach using the zeta function. Understanding the new function created will provide insight into the frequency of prime numbers.

Throughout this paper, we will establish a comprehensive theoretical foundation and rigorously develop the methodology to investigate the structure of quotient ring S/P under the influence of symmetric generalized n-derivations, where S represents an arbitrary ring, and P signifies a prime ideal of S satisfying certain algebraic identities acting on prime ideal P without relying on the assumption of primeness or semi-primeness of the ring.

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the *-version of Vukman’s theorem, we show that if a 2-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.

The Ramanujan primes are the least positive integers Rn having the property that if m ≥ Rn, then π_m − π_(m/2) ≥ n. This document develops several bounds related to the Ramanujan primes, sharpening the currently known results. The theory presented is by no means exhaustive, however it provides insights for future research work. Alternatively, we may say that it is a road map which may be followed to make further discoveries.

Twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the part “twin prime” is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid proof for twin prime conjecture.

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\pi(x)$ denotes the \textit{Prime Counting Function}. The proofs require specific order estimates involving $\pi(x)$ and the \textit{Second Chebyshev Function} $\psi(x)$, as well as the famous \textit{Prime Number Theorem} in addition to certain meromorphic properties of the \textit{Riemann Zeta Function} $\zeta(s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.

In this paper, we propose a novel stochastic model to study prime distribution in almost-short intervals, inspired by a corollary of the Maynard-Guth theorem. The model uses the equation \[ x_{n+1} = x_n + \frac{y}{\log x_n} + \epsilon_n, \] where \(\epsilon_n\) follows a Laplace distribution, and demonstrates that our new bound surpasses both Walfisz and Maynard-Guth bounds, particularly for large \(x\): \[ \left| \pi(x + y) - \pi(x) - \frac{y}{\log x} \right| \leq C \cdot y \exp(-\sqrt[4]{\log x}). \] Additionally, we explore a modulated Hamiltonian \(\hat{H}(x)\), derived from a potential \(V(x)\) and a test function \(f(x)\), that exhibits behavior akin to the energy levels of the heavy nucleus \(H_{38}\). The gaps between these energy levels resemble the distribution of zeros of the Riemann zeta function near the critical line, suggesting that \(\zeta(0.5 + i \hat{H}(x))\) corresponds to the energy levels of \(\hat{H}(x)\), thus supporting a quantum mechanical interpretation of the Riemann hypothesis. Furthermore, assuming the Elliott-Halberstam conjecture, which implies that prime gaps can be as small as 12, we find that the transition rate \(\Gamma_{p \to p+12}\) between quantum states associated with prime numbers increases with larger primes. This observation indicates that the distribution of prime gaps reflects an underlying discrete structure in the "energy" landscape of primes, offering a novel perspective on the connection between number theory and quantum physics.

The present study aims to show on the one hand how easily one can get confused when doing research on the internet. Different platforms, fora and homepages supply the visitor with different - and sometimes contradictory - information. On the other hand, we show how to overcome these contradictions and produce valuable research based on almost all sources. For our study, we use the example of the Feit-Thompson conjecture, which is - in opposite to the well-known Feit-Thompson theorem - not widely known. We try to clarify the scientific literature on this topic and we embed our own result into the present state of art. We prove that pq−1p−1 does not divide qp−1q−1 if p and q are twin primes, i.e. p=q+2. Thus we prove that the Feit-Thompson conjecture is true for twin prime numbers.

Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof for the equation $a^n + b^n = c^n$ having no positive integer solutions for exponents $n$ greater than 2. This statement, now known as Fermat's Last Theorem, remained unproven for centuries, despite the efforts of countless mathematicians. Andrew Wiles' work in 1994 finally provided a rigorous proof of Fermat's Last Theorem. However, Wiles' proof relied on advanced mathematical techniques that were far beyond the scope of Fermat's time, raising questions about whether Fermat could have truly possessed a proof using the methods available to him. Wiles's achievement was widely celebrated, and he was awarded the Abel Prize in 2016 in recognition of his groundbreaking work. The citation for the award described his proof as a "stunning advance" in mathematics. The present work offers a potential solution to Fermat's Last Theorem that may be more aligned with the original approach that Fermat claimed to have used.

The Diophantine equation for which p is prime 2^n = p^2 + 7 and n is integers with p > 0, is examined in this work. It is demonstrated by examining the characteristics of the equation that the only solutions meeting these requirements are p = 3 with n = 5 and p = 5 with n = 7. In order to analyze the above equation, it is necessary to look at the properties of primes and how they relate to powers of two. These results advance our knowledge of prime numbers and how they behave in certain mathematical problems.

The Goldbach Conjecture, which is frequently termed as “1 + 1”, has been a fascinating goal for many mathematicians over centuries. A Chinese mathematician, Dr. Jingrun Chen, proved 1 + 2, which is a great success and the best result so far achieved. Although there were several attempts proving the conjecture, these attempts are either tediously long, complicated, or logically imperfect, thus not widely accepted. Taking advantage of the periodicity of primes, here the author provides a straight forward rigorous proof for the Conjecture.

Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.

A twin prime numbers are two prime numbers which have the difference of 2 exactly. In other words, twin primes is a pair of prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Up to date there is no any valid proof/disproof for twin prime conjecture. Through this research paper, my attempt is to provide a valid disproof for twin prime conjecture.

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

A prime number is a natural number with no divisors other than itself and the number 1. There are many unsolved problems related to prime numbers. One such problem is finding a general method for identifying prime numbers. Although it is known that all prime numbers fall under the categories of 2, 3, or 6n±1 (where n≥1), a formalized method for eliminating composite numbers (non-prime numbers) from these categories has yet to be developed. This paper aims to explore a possible method for such elimination, which involves checking whether there exists an n that satisfies n = 6km+k+m or n = 6km+k-m (where k and m are non-zero integers). This method is expected to be a major step forward in the derivation of prime numbers.

Bacillus Calmette-Guérin (BCG) is the only widely used prophylactic tuberculosis (TB) vaccine that can prevent severe TB in infants. However, it provides poor protection in adults, and therefore, there in ongoing research on new TB vaccines and immunization strategies with more durable immune effects. The recombinant BCG and BCG prime-protein booster are two important vaccine strategies that have recently been developed based on BCG and could improve immune responses. In this study, three immune strategies based on four protective antigens, namely, ESAT-6, CFP-10, nPPE18, and nPstS1, were applied to construct recombinant rBCG-EPCP009, EPCP009 subunit protein, and BCG prime-EPCP009 booster vaccine candidates. The short- and long-term immune effects after vaccination in Balb/c mice were evaluated based on humoral immunity, cellular immunity, and the ability of spleen cells to inhibit in vitro mycobacterial growth. At 8 and 12 weeks after the initial immunization, splenocytes from mice inoculated with the BCG prime-EPCP009 protein booster secreted higher levels of PPD- and EPCP009-specific IFN-γ, IL-2, TNF-α, IL-17, GM-CSF, and IL-12 and had a higher IFN-γ⁺CD4⁺ TEM:IL-2⁺CD8⁺ TCM cell ratio than splenocytes from mice inoculated with the rBCG-EPCP009 and EPCP009 proteins. In addition, the EPCPE009-specific IgG2a/IgG1 ratio was slightly higher in the BCG prime-EPCP009 protein booster group than in the other two groups. The in vitro mycobacterial inhibition assay showed that the splenocytes of mice from the BCG prime-EPCP009 protein booster group exhibited stronger inhibition of Mycobacterium tuberculosis （M. tuberculosis）growth than the splenocytes of mice from the other two groups. These results indicate that the BCG prime-EPCP009 protein booster exhibited superior immunogenicity and M. tuberculosis growth inhibition to the parental BCG, rBCG-EPCP009, and EPCP009 proteins under in vitro conditions. Thus, the BCG prime-EPCP009 protein booster may be important for the development of a more effective adult TB vaccine.

In this study, we define the concept of pseudo-prime subhypermodules of hypermodules as a generalization of the prime hyperideal of commutative hyperrings in [17]. Firstly we examine the spectrum of the Zariski topology, which we built on the element of the pseudo-prime subhypermodules of a class of hyper-modules, then we give some relevant properties of the hypermodule in this topological hyperspace.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in Γ(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G)=Γ(P), G has a composition factor isomorphic to P. In [4] proved finite simple groups ²D_n(q), where n ≠ 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups ²D_2k(q), where k ≥ 9 and q is a prime power less than 10⁵.

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

In this paper, a "local" algorithm is determined for the construction of two recurrent sequences of positive primes ( ) and ( ), (( ) dependent of ( ) ), such that for each integer n their sum is equal to 2n . To form this, a third sequence of primes ( ) is defined for any integer n by : = Sup( p ∈ : p ≤ 2n - 3 ) , where is the infinite set of primes. The Goldbach conjecture has been proved for all even integers 2n between 4 and 4. In the table of terms of Goldbach sequences given in appendix 10 , values of the order of 2n = are reached. This " finite ascent and descent " method proves the binary Goldbach conjecture ; an analogous proof by recurrence is established and an increase in by 0.7( is justified. Moreover, the Lagrange-Lemoine-Levy conjecture and its generalization, the Bezout-Goldbach conjecture, are proven by the same type of procedure.

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p · n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p · n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 · n · p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 · n · p + p formula we put forth with another known formula.

This paper presents proof of Riemann's hypothesis by utilizing the golden ratio and the ellipse's eccentricity in six dimensions. This proof extends across all scientific fields. this research introduces logical and mathematical principles for distributing, recognizing, and categorizing prime numbers. It also explores the relationship between prime numbers and the complex conjugate of five prime numbers in various dimensions. Accordingly, it delves into the classification of even numbers based on prime numbers. This study examines the proof of Riemann's hypothesis from a geometric dimension perspective. Based on this, the mathematical structure of an integrated theory is defined. properties and principles associated with prime numbers reveal connections between quantum mechanics and general relativity.

The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma} \cdot n \cdot \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. In this note, using Robin's criterion on superabundant numbers, we prove that the Riemann hypothesis is true.

When discussing the Diophantine quintic equation $p(a^5 + b^5) = q(c^5 + d^5)$ where p is a prime and q is an integer, there is a clear gap between the mathematical literature and online forums. Because of the intrinsic complexity of parameterizing fifth-degree equations, this equation re- mains largely unexplored. In this work, we approach this quintic problem through algebraic methods with the goal of providing numerical solutions that illuminate its properties and behaviors.Our research uncovers in- triguing patterns and connections that shed light on the enigmatic nature of Diophantine quintic equations in this particular form.

One of the most important unresolved mysteries in mathematics is the Riemann Hypothesis, which suggests a fundamental connection between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers. Here, we explore the fascinating union of trigonometric functions, prime numbers, and the Riemann zeta function through an examination of the zeros in the statement ln(sec(π·n log(n))). We demonstrate a strong mathematical connection between these components, providing information on the mysterious properties of prime numbers and their complex relationships to basic mathematical operations. Our thorough investigation adds to the current discussion of the Riemann Hypothesis by offering possible solutions and deepening our comprehension of the intricate relationship between number theory and analytic functions.

This paper serves as a seamless continuation of the previous work titled "Special Spirals are Produced by the ROTASE Galactic Spiral Equations with the Sequential Prime Numbers." Upon revisiting the data, a noteworthy observation emerged: the special spirals can be meticulously aligned with the prime spiral on a point-by-point basis through careful scaling and rotation for the parameter ρ remaining a constant greater than one. As ρ increases, both the scaling factor and rotation angle exhibit a corresponding augmentation, albeit with an intriguing caveat—the rotation angle asymptotically approaches its limit of 90°. Notably, the galactic spiral can be perceived as a remarkably close approximation of the Archimedean spiral, with the degree of proximity increasing as ρ grows larger. Furthermore, an intriguing correspondence arises when ρ takes on a value of 1; at this value, the galactic spiral bears a striking resemblance to the Fermat spiral, the space between two consecutive spiral loops undergoes a continuous reduction as the loop's radius expands, yet it never diminishes to zero; simultaneously, the area between two successive loops experiences an augmentation with the value of ρ.

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann hypothesis should be false. The Riemann hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\frac{3 \times \log x + 5}{8 \times \pi \times \sqrt{x} + 1.2 \times \log x + 2} + \frac{\log x}{\log \theta(x)} \leq 1$ is satisfied for some number $x \geq 13.1$.

Let {$\Gamma(k)$} denote {$(k-1)!$}, and let {$\Gamma_{n}(k)$} denote {$(k-1)!$}, where {$n \in \{3,\ldots,16\}$} and {$k \in \{2\} \cup [2^{\textstyle 2^{n-3}}+1,\infty) \cap \mathbb N$}. For an integer {$n \in \{3,\ldots,16\}$}, let $\Sigma_n$ denote the following statement: if a system of equations {${\mathcal S} \subseteq \{\Gamma_{n}(x_i)=x_k:~i,k \in \{1,\ldots,n\}\} \cup \{x_i \cdot x_j=x_k:~i,j,k \in \{1,\ldots,n\}\}$} with $\Gamma$ instead of $\Gamma_{n}$ has only finitely many solutions in positive integers {$x_1,\ldots,x_n$}, then every tuple {$(x_1,\ldots,x_n) \in (\mathbb N \setminus \{0\})^n$} that solves the original system ${\mathcal S}$ satisfies {$x_1,\ldots,x_n \leqslant 2^{\textstyle 2^{n-2}}$}. Our hypothesis claims that the statements {$\Sigma_{3},\ldots,\Sigma_{16}$} are true. The statement {$\Sigma_6$} proves the following implication: if the equation {$x(x+1)=y!$} has only finitely many solutions in positive integers $x$ and $y$, then each such solution {$(x,y)$} belongs to the set {$\{(1,2),(2,3)\}$}. The statement {$\Sigma_6$} proves the following implication: if the equation {$x!+1=y^2$} has only finitely many solutions in positive integers $x$ and $y$, then each such solution {$(x,y)$} belongs to the set {$\{(4,5),(5,11),(7,71)\}$}. The statement {$\Sigma_9$} implies the infinitude of primes of the form {$n^2+1$}. The statement {$\Sigma_9$} implies that any prime of the form {$n!+1$} with {$n \geqslant 2^{\textstyle 2^{9-3}}$} proves the infinitude of primes of the form {$n!+1$}. The statement {$\Sigma_{14}$} implies the infinitude of twin primes. The statement {$\Sigma_{16}$} implies the infinitude of Sophie Germain primes.

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

In this paper, the distribution of prime numbers is expressed based on proving the Riemann hypothesis. The relationship between three numbers, three, six, and nine, and the modality to the distribution of prime numbers, is one of the results of Riemann's zeta function. Prime numbers are classified into six groups of single-digit numbers. There are no prime numbers in groups of three, six, and nine. The groups are made based on the sum of the internal digits. And for each set, there is an angle in the complex plane. The distance between the prime numbers in each group has a regular pattern. This pattern is a multiple of the numbers three, six, and nine. According to Euler's number, for an angle of 60 degrees, the real part of the cosine is 1/2. Accordingly, all prime numbers are related to angles greater than 60 degrees to 90 degrees. As a result, based on the relationship between the golden spiral and the complex conjugate of the zeta function, the function in The 1/2 point becomes zero based on the prime numbers.

This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.

Sialyllactose (SL) is a functional human milk oligosaccharide essential for immune support, brain development, intestinal maturation, and antiviral defense. However, despite its established health benefits, the effect of SL on exercise performance and muscle mass in mice remains unknown. Here, we aimed to investigate, for the first time, the effects of 6′-SL on muscle functions. Seven-week-old male C57BL/6J mice were administered 100 mg/kg 6′-SL for 12 weeks, after which exhaustive treadmill performance, muscle strength, and muscle phenotype were examined. The administration of 6′-SL significantly improved exhaustive treadmill performance metrics, including distance and exhaustion time. Grip strength was also increased by 6′-SL administration. Additionally, 6′-SL increased muscle mass in both the gastrocnemius (GAS) and soleus. 6′-SL administration led to an increase in the minimum Feret’s diameter and the protein expression of total myosin heavy chain in the GAS muscle. In conclusion, 6′-SL administration in vivo led to increased running distance and time by increasing muscle mass and strength. These findings collectively indicate that 6′-SL is a potential agent for improving muscle health and exercise performance.

Genetic engineering has become an essential element in developing climate-resilient crops and environmental sustainable solutions to respond to the increasing need for global food security. Genome editing using CRISPR/Cas [Clustered regulatory interspaced short palindromic repeat (CRISPR)-associated protein (Cas)] technology is being applied to a variety of organisms, including plants. This technique has become popular because of its high specificity, effectiveness, and low production cost. Therefore, this technology has the potential to revolutionize agriculture and contribute to global food security. Over the past few years, increasing efforts have been seen in its application in developing higher-yielding, nutrition rich, disease resistant, stress-tolerant “crops” fruits and vegetables. Cas proteins such as Cas9, Cas12, Cas13, and Cas14, among others, have distinct architectures and have been used to create new genetic tools that improve features that are important for agriculture. The versatility of Cas has accelerated genomic analysis and facilitated the use of CRISPR/Cas to manipulate and alter nucleic acid sequences in cells of different organisms. This review provides the evolution of CRISPR technology exploring its mechanisms and contrasting it with traditional breeding and transgenic approaches to improve different stress tolerance. We have also discussed the CRISPR/Cas system and explored three Cas proteins that are currently known to exist: Cas12, Cas13, and Cas14 and their potential to generate foreign DNA free or non-transgenic crops that could be easily regulated for commercialization in most countries.

It is proved that there exist two odd prime numbers, p_i and p_j , such that p_i + p_j = 2r + 2(r ⩾ 2), where p_n denote the n-th odd prime number, 1 ⩽ i, j ⩽ n. By the theory of generating functions, we prove that there exist two positive integers, h_i and h_j , such that h_i + h_j = r(r ⩾ 2), where h_n = (p_n − 1)/2，(p_n ⩾ 3).Then we complete the proof.

Prime numbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advances in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In that paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function defined by: Where ζ(z) is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of the zeros of RZF, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this: [RH] The real part of every nontrivial zero z* of the RZF is 1/2. Proving the RH is, as of today, one of the most important problems in mathematics. In this paper we will provide a proof of the RH. The proof of the RH will be built following these five parts: PART 1: Description of the RZF PART 2: The C-transformation PART 3: Application of the C-transformation to in Re(z) 0 to obtain ζ(z)=X(z)-Y(z) PART 4: Analysis of the values of z such that X(z)=Y(z), and |X(z)|=|Y(z)|, that equates to ζ(z)=0 Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 Conclude that ζ(z)=0 only if Re(z)=1/2 for Re(z) 0 PART 5: We will also prove that all non-trivial zeros of ζ(z) in the critical line of the form are not distributed randomly. There is a relationship between the values of those zeros and the Harmonic function that leads to an algebraic relationship between any two zeros.

In this study, a simple approach to solving the Riemann Hypothesis, one of the most prominent unsolved problems in the field of Number Theory in mathematics and one of the “Millenium Prize Problems”, is presented. The Riemann Hypothesis, a conjecture about distribution of prime numbers, has remained unsolved since it was first proposed by German mathematician, Bernhard Riemann in 1859. This paper introduces an analytic continuation of the Zeta Function as well as symmetric approach through integration by parts to address this hypothesis. The proposed solution is obtained through analytical continuation in the critical strip 0<Re(s)<1 to establish that Re(s)=1/2.

For each non-prime odd number as F=pq , if we consider m/n as an approximation for q/p and choose k=mn , then by proving some lemmas and theorems, we can compute the values of m and n. Finally, by using Fermat’s factorization method for F and 4kF as difference of two non-consecutive natural numbers, we should be able to find the values of p and q. Then we introduce two new and powerful sieves for separating composite numbers from prime numbers.

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

This paper definitively settles the longstanding conjecture regarding odd perfect numbers. A perfect number is one whose sum of divisors equals itself doubled. While Euclid's method for constructing even perfect numbers is well-known, the existence of odd ones has remained elusive. By employing elementary techniques and analyzing the properties of the divisor sum function, we conclusively prove that no odd perfect numbers exist.

Prime numbers are the building blocks of number theory, and their properties have intrigued mathematicians for centuries. In this paper, we explore a specific categorization of primes, which we refer to as "twin primes", “cousin primes” and "sexy primes". We begin by examining the form of prime numbers greater than 7 and analyze their potential to be expressed in terms of $6n + 5$ or $6n + 7$. This investigation leads to the identification of conditions under which prime numbers can form twin, cousin and sexy pairs. We further develop a general framework to characterize the product of all twin (except 15), cousin (except 21) and sexy primes, providing insights into their algebraic structure.

We demonstrate a new quantitative method to the sieve of Eratosthenes, which is an alternative to the sieve of Legendre. In this method, every element of a given set is sifted out once only, and therefore, this method is free of the Mobius function and of the parity barrier. Using this method, we prove that every sufficiently large even number is the sum of two primes, and that every even number is the difference of two primes in infinitely many ways.

Integer is either a composite number or a prime number. Therefore, detecting composite numbers is important for solving prime numbers. The study of prime numbers, apart from satisfying human curiosity, can be very important. In this article, the order of composite numbers has been detected. And explained with a simple method and a simple function. And, a method has been developed in which all composite numbers and therefore prime numbers can be determined by using the specified methods, functions and formulas.

The paper presents two novel mathematical approaches to prove Goldbach's conjecture. The conjecture states that every even number greater than or equal to 6 can be expressed as the sum of two odd prime numbers. We propose that the conjecture has only four genetic codes, and as long as one of them remains valid, the conjecture continues to hold. Through these new mathematical ideas, we aim to prove the validity of Goldbach's conjecture comprehensively.

The Riemann hypothesis, renowned for its deep connection to the distribution of prime numbers, remains a central problem in mathematics. Understanding the distribution of primes is crucial for developing efficient algorithms and advancing our knowledge of number theory. The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. Several equivalent formulations of the Riemann hypothesis exist. Robin's criterion for the Riemann hypothesis is based on an inequality that divisor sum function $\sigma$ must satisfy at natural numbers greater than 5040. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. By using Robin's criterion on superabundant numbers, we present a novel approach that culminates in a complete proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.

In the article, we define a sequence F: F(0) = 2, F(1) = 10, F(2) = 16, F(3) = 22, F(4) = 26, F(4) = 28, · · · , F(i) = F(i − 1) + l, where the number l ≤ 8 is defined by using prime numbers and matrix algebra. The sequence includes only even numbers and can be used to define a series of numbers F(i) + 1, the non-Goldbach sequence, that contains all numbers that cannot be written as the sum of two primes. The result is related to Goldbach’s conjecture. The famous Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.

Based on recent popular money priming research results, which conclude that money makes self-sufficient (e.g. less interest in other people), we assumed that people are less interested in finding out whether others are lying or telling the truth. In a laboratory experiment, 163 students (85 women, 78 men, M_Age = 23.08, ranging from 18 to 36 years) were primed by actively handling money (versus paper sheets). Afterwards, they classified 24 video statements as true or deceptive (senders describing their most/least favorite movie), rated their classification confidence for each decision and then answered control questions. Results revealed no influence of priming condition on judgmental bias, classification accuracy, and classification confidence. Also the level of self-reported motivation to find out who lied or told the truth did not differ between conditions. Higher motivation was correlated to higher classification confidence. Additionally, and in line with Reinhard (2010) and Reinhard et al. (2011), higher classification accuracy correlated to a higher use of verbal content cues for classification decisions. Thus, while we were able to replicate these findings, our results contradict the assumption of a money prime influence on lie detection ability. Concluding, our results make self-sufficiency in this context questionable and offer next steps for research.

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P₁ = 1 + 6 * n (n > 0) P₅ = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P₁ prime number or a P₅ prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P₁ prime numbers roughly equal the number of P₅ prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.

VITEK MS PRIME (bioMerieux), a newly developed Matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF MS) system, alongside VITEK PICKME Pen (PICKME), offers easy sample preparation for bacteria and yeast. VITEK MS PRIME also offers two software platforms for filamentous fungi: the IVD database and the RUO database. Our study evaluated its identification agreement on 320 clinical isolates of bacteria and yeasts, comparing PICKME and traditional wooden toothpick sampling techniques against ASTA Mi-croIDSys Elite results. Additionally, we assessed the IVD (v3.2) and SARAMIS (v4.16) RUO data-bases on 289 filamentous fungi against molecular sequencing. The concordant rates for spe-cies-level identification of bacteria and yeasts were about 89.4% (286/320) between the PICKME and wooden toothpick, about 83.4-85.3% between VITEK MS PRIME and ASTA MicroIDSys Elite. Retesting with PICKME improved concordance to 91.9%. For filamentous fungi, species-level identification reached 71.3% with the IVD database and 85.8% with RUO, which significantly enhanced basidiomycetes' identification from 35.3% to 100%. Some strains in the IVD database, like Aspergillus vesicolor, Exophiala xenobiotica, and Nannizzia gypsea failed to be identified. VITEK MS PRIME with PICKME offers reliable and efficient microorganism identification. For fila-mentous fungi, combining the use of the RUO database can be beneficial, especially for basidi-omycetes.

In this paper for every number field K generated by a root α of a trinomial x⁷ + ax + b ∈ Z[x] and for every prime integer p, we calculate ν_p(i(K)), the highest power of p dividing the index i(K) of the field K. In particular, we calculate the index i(K). As application, when the index of K is not trivial, then K is not monogenic.

Adeno-associated viral (AAV) vectors can be used for gene delivery. AAV.CAP-Mac was recently developed; it can cross the blood-brain barrier and transduce cells throughout the CNS. However, some brain regions were not transduced in adult rhesus monkeys. Additionally, AAV vectors can only package 5 kb of DNA. Also, AAV vectors may be genotoxic and cytotoxic, especially at high doses. Finally, AAV vectors are very expensive to produce at sufficiently high titers for treatment. Off-the-shelf cell-based delivery systems wherein the cells can infiltrate the target tissue and be hyper-motile would be ideal. Also, mRNA-based gene editing would be a transient treatment, and thus be much safer.

Let R be a commutative ring with identity and m, n be positive integers. In this paper, we introduce the class of (m,n)-prime ideals which lies properly between the classes of prime and (m,n)-closed ideals. A proper ideal I of R is called (m,n)-prime if for a,b∈R, a^{m}b∈I implies either aⁿ∈I or b∈I. Several characterizations of this new class with many examples are given. Analougus to primary decomposition, we define the (m,n)-decomposition of ideals and show that every ideal in an n-Noetherian ring has an (m,n)-decomposition. Furthermore, the (m,n)-prime avoidance theorem is proved.

Abiotic stresses, including drought, salinity, cold, heat, and heavy metals, extensively reduce global agricultural production. Approaches such as conventional breeding and transgenic breeding have been widely used to cope with these environmental stresses. The clustered regularly interspaced short palindromic repeat- Cas (CRISPR/Cas) based gene-editing tool has revolutionized due to its simplicity, accessibility, adaptability, flexibility, and wide applicability. This system has a great potential to build up crop varieties with enhanced tolerance against abiotic stresses. In this review, we summarize the most recent findings on understanding the mechanism of abiotic stress response in plants and the application of CRISPR/Cas mediated gene-editing system towards enhanced tolerance to drought, salinity, cold, heat, and heavy metals stresses. Furthermore, in this review, we highlighted the recent advancements in prime editing and base editing tools for crop improvement.

Therapeutic vaccine studies should be designed to elicit durable, high magnitude, and efficacious T cell responses, all of which can be impacted by the choice of vaccination schedule. Here, we compare different prime-boost intervals (PBI) in a human papillomavirus (HPV) model using HPV16E6E7 Venezuelan equine encephalitis virus replicon particle (VRP) vaccination to address the optimal boosting schedule, quality of immune response, and overall in vivo efficacy. Six different vaccine regimens were tested with each group receiving booster vaccinations at different time intervals. Analysis of T-cell responses demonstrated a significant HPV16 E7 specific CD8+T cell response with minimally a one-week PBI between antigen re-exposure. Significant E7-specific in vivo cytotoxicity was also observed with longer PBIs. Additionally, longer PBIs led to an enhanced memory recall response to tumor challenge, which correlated with differential expansion of T cell memory subsets. Our findings imply that when using alphavirus vector platforms as a vaccination strategy, a one-week PBI is sufficient to induce high magnitude effector T cells with potent anti-tumor activity. However, longer PBIs lead to enhanced long-term protective anti-tumor immunity. These findings have implications for therapeutic vaccine clinical trials in which shorter intervals of prime-boost regimens may lead to suboptimal durable immune responses.

This paper brings to fruition recently found \cite{migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of finding the statistical limit of the Euler ensemble. We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the \textbf{quantum} statistical theory of $N$ fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the turbulent limit $N\to \infty, \nu \to 0$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the density of these fermions. This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without approximations or fitted parameters. The energy spectrum decays as $k^{-\frac{7}{2}}$. We compute resulting integrals in \Mathematica{} and present effective index $n(t) = -t\partial_t \log E $ for the energy decay as a function of time Fig.\ref{fig::NPlot}. The asymptotic value of the effective index in energy decay $n(\infty) = \frac{5}{4}$, but the universal function $n(t)$ is neither constant nor linear.

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $\sigma_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum \sigma_i = q r$. This equivalence provides a \textbf{dual} theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a \textbf{quantum} statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $\mu \rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $\mu \rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $\mu$ using methods of number theory. It grows as $\nu/\mu^2$ in the continuum limit $\mu \rightarrow 0$, leading to anomalous dissipation at $\mu \propto \sqrt{\nu} \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-\lambda}$, where the anomalous dimensions $\lambda$ satisfy the spectral equation \eqref{spectralEq}. The spectrum becomes a continuum in the statistical limit $N \to \infty$, presumably leading to multifractal phenomena.

In this paper, we give explicit asymptotic formulas for some sums over primes involving generalized alternating hyperharmonic numbers of types I, II and III. Analogous results for numbers with $k$-prime factors will also be considered.

Sickle cell disease is characterized by stiff, “sickled” red blood cells that have difficulty moving through the bloodstream and do not efficiently carry oxygen. It is an inherited disease with severely limited treatment options, and is caused by a point mutation. Its prevalence in black and brown communities makes the already limited treatment options even less accessible. Base editing and prime editing are two relatively recent discoveries in the field of genome editing and were developed after the groundbreaking discovery of the CRISPR Cas9 system. While not fully tested, they hold a lot of promise in providing alternative treatment options for sickle cell disease. Both editing systems are able to install individual point mutations in the beta globin gene, which is where the sickle cell mutation occurs, and can thus cure sickle cell disease (in theory). In this paper we outline the mechanisms of CRISPR-Cas9 systems and base and prime editing, and provide insight into how to apply them to treat SCD. Further investigation should be done on specific editing systems and designs to use to ensure optimal treatment of SCD.

Let $R$ be a commutative ring with identity. A proper ideal $I$ of a ring $R$ is called a square-difference factor absorbing primary ideal of $R$ if for $a,b\in R$, whenever $a^{2}-b^{2}\in I$, then $a+b\in\sqrt{I}$ or $a-b\in I$. Several characterizations and properties of this class of ideals are presented.Various examples are provided to illustrate the obtained results and demonstrate the applicability of our findings. Furthermore, the properties of this class of ideals are investigated in extensions of rings.

Let R be a commutative ring and m, n be positive integers. We define a proper submodule N of an R-module M to be (m,n)-closed if for r∈R and b∈M, r^{m}b∈N implies rⁿ∈(N:_{R}M) or b∈N. This class of submodules lies properly between the classes of prime and primary submodules. Many characterizations, properties and supporting examples concerning this class of submodules are provided. The notion of (m,n)-modules is introduced and characterized. Furhermore, the (m,n)-closed avoidance theorem is proved. Finally, the (m,n)-closed submodules in amalgamated modules are studied.

Conventional plant breeding methods exploit already existing genomic variation in plants to develop a variety in 8 to 10 years, which can decrease the genetic variability of the plant’s genome. The ever-increasing food demand for cereals crops cannot be met by traditional breeding methods. In order to increase food production in less time, there is a dire need to improve breeding methods. Several conventional and molecular breeding methods are being used to improve the crops traits. Molecular researchers have developed new genome editing tools like CRISPR/Cas9, CRISPR/Cpf1, prime editing, base editing, dcas9 epigenetic modification, and several other transgene-free genomes editing approaches. These genome editing tools can improve the desired traits precisely and efficiently. Moreover, a newly developed breeding method “Speed Breeding” has revolutionized the agriculture by shortening the crop cycle. It can produce 5-6 generations of cereals in a year. In this review, we have summarized all these conventional and molecular breeding approaches to improve cereal crops.

This study investigated live animal performance and carcass characteristics of Australian prime lambs fed oil based polyunsaturated fatty acid (PUFA) enriched pellets in a feedlot system. The tested hypothesis was that supplementation of lambs with a variety of dietary oil based PUFA enriched pellets would enhance growth and carcass characteristics compared with the control lambs on lucerne only. Seventy-two, 6 months old White Suffolk x Corriedale first-cross prime lambs with an average liveweight (LWT) of 35.7 ± 0.9 kg were allocated to six treatment groups in a completely randomised experimental design. The treatments were: (1) control: lucerne hay only; or lucerne hay plus wheat-based pellets infused with 50 ml/kg DM of oils from (2) rice bran (RBO); (3) canola (CO); (4) rumen protected (RPO); (5) flaxseed (FO) and (6) safflower (SO) dietary sources. All lambs had ad libitum access to lucerne hay and clean fresh water. Supplemented lambs were fed 1kg of pellet/head/day for 10 weeks. Feed intake, final LWT, average daily gain (ADG), body conformation and carcass characteristics of lambs in the supplemented groups were all greater than for the control group. SO lambs had the lowest ADG of 190.3 g/day. RBO and CO treatments had the lowest feed cost per unit gain of AU$ 3.0/kg. Supplemented lambs had similar over the hooks (OTH) incomes that were all higher than that of the control group. This empirical evidence-based data demonstrated that supplementation of lambs with RBO and CO had comparatively lower feed costs without compromising ADG, carcass characteristics and OTH income.

Viruses are silent enemies that intrude and take control of the plant cell's machinery for their own multiplication. Infection by viruses and the resulting damage is still a major challenge in the agriculture sector. Plants have the ability to fight back, but the ability of viruses to counteract and mutate at a fast rate helps them to hide from the host response. Therefore, classical approaches for introgressing resistance genes by breeding have obtained limited success in counteracting the virus menace. Genetic modification (GM) based strategies have been successful in engineering artificial resistance in plants. Several different approaches based on pathogen derived resistance, antisense constructs, hairpin RNA, double stranded RNA etc. have been used to enhance plant resistance to the virus. Recently, genome editing (GE) strategies mainly involving the CRISPR/Cas mediated modifications are being used for virus control. In this review, we discuss the developments and advancements in the GM and GE based methods in tackling virus infection in plants.

We report the first correction by Prime editing of a mutation in the RYR1 gene, paving the way to gene therapies for RYR1-related myopathies. The RYR1 gene codes for a calcium channel named Ryanodine receptor 1, which is expressed in skeletal muscle fibers. The failure of this channel causes muscle weakness in patients, which degenerates into motor disabilities. Currently, there are no effective treatments for these diseases, which are mainly caused by point mutations. Prime editing allows the modification of precise nucleotides in the DNA. Our results showed a 59% correction rate of the T4709M mutation in the RYR1 gene in human myoblasts by RNA delivery of the Prime editing components. It is to be noted that the T4709M is recessive, and thus, persons having a heterozygous mutation are healthy. These results are the first demonstration that it is possible to correct mutations in the RYR1 gene.

This Paper intends to introduce the notion of prime ideals of Intuitionistic fuzzy normed Rings and to establish basic properties related to it. It investigates these notions and shown anew Result using intuinistic fuzzy points and non membership function incorparating with t-norm and s-norm to show aome results of fuzzy prime ideal.

The RNA-guided endonuclease system CRISPR-Cas9 has been extensively modified since its discovery, allowing its capabilities to be extending far beyond double-stranded cleavage to high fidelity insertions, deletions, and single base edits. Such innovations have been possible due to the modular architecture of CRISPR-Cas9 and the robustness of its component parts to modifications and the fusion of new functional elements. Here, we review the broad toolkit of CRISPR-Cas9-based systems now available for diverse genome editing tasks. We provide an overview of their core molecular structure and mechanism and distil the design principles used to engineer their diverse functionalities. We end by looking beyond the biochemistry and towards the societal and ethical challenges that these CRISPR-Cas9 systems face if their transformative capabilities are to be deployed in a safe and acceptable manner.

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s zeta diverges. Dirichlet’s eta function η(s), negates alternate terms of zeta, permitting convergence when s∈C and Re(s) < 1, and its non-trivial zeros {ρ}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(ρ) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(s) ≠ 1/2. Analytical continuation generates a modified eta series η_l(s), in which every l^th term is multiplied by (1-l). If the integer l is appropriately determined by the Im(s), similar paired finite vector series have a difference that closely follows η_l (s) and their terminal vectors intersect in a unique way permitting zeros only when Re(s) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(s) > 1/2.

Statistical modeling of lifetime data plays an essential role in a wide range of practical fields, such as health and engineering. There have been a lot of studies done to develop statistical models that can better describe health data than traditional models. For the first time, we pioneer a novel family of continuous probability distributions called the generalized odd beta prime generalized (GOBP-G) family of distributions. The cumulative distribution and probability density functions of the new family are presented. A new generalization of the Weibull distribution called "generalized odd beta prime-Weibull" (GOBPW) is proposed using the pioneered GOBP-G family. The mixture representations of the new distribution are defined and derived. Some formal statistical properties of the GOBPW distribution, such as the moments, moment generating function, incomplete moments, information generating function, entropies, stress-strength function, quantile function, and order statistics, are derived. The estimation of the parameters of the proposed distribution is evaluated using the maximum likelihood estimation approach. Different cancer disease data sets, such as the bladder, head and neck, acute bone, and blood cancers, are used to illustrate the applicability and usefulness of the new model and were compared using several statistical accuracy measures with that of well-established extended Weibull distributions, which are the beta modified Weibull distribution, Kumaraswamy modified Weibull distribution, gamma generalized modified Weibull distribution, gamma log-logistic Weibull distribution, and beta log-logistic Weibull distribution. The results show that the proposed model gives better results than the competitive models. This study could guide the relevant stakeholders in choosing a suitable statistical model for the health data instead of relying on traditional models to enhance decision-making.

One of the rare diseases throughout the world is Glycogen Storage Disease, which appears due to problems in glycogen metabolism. Among various subtypes of GSD, GSD Type 1a is the most abundant one of GSD Type 1, seen in approximately 80% and caused by different kinds of mutations in the Glucose-6-Phosphatase Catalytic Subunit (G6PC) gene in human chromosome 17q21. G6PC gene encodes for glucose-6-phosphatase (G6Pase) protein, which cleaves glucose-6-phosphate into glucose and inorganic phosphate (Pi), and GSD Type 1a patients fail to breakdown glucose-6-phosphate due to several mutations in the G6PC gene. In our study, we aim to create new therapeutic approaches for GSD 1a. We collected mutation data of 57 GSD Type 1a patients from Turkey. According to the data, 16 types of mutations were observed in the G6PC gene. Allele frequencies of these mutations are calculated as 59% for R83C/H, 11% for W160*, 7% for G270V, and 28% for others which have less frequency. Up to now, the tertiary protein structure of G6Pase has not been structured yet. To understand the possible impacts of these mutations, we statistically obtained possible tertiary structure predictions of G6Pase by running 5 different tools. At the end of the study, we suggest two effective and promising gene therapy methods for GSD Type 1a, Prime Editing for R83C/H mutations, and mRNA delivery for other mutations, in addition to a promising, commercially available drug suggestion for patients with W160*, W86*, and S15* mutations, although the drug belongs to another disease.

This paper presents a new proof of the infinite nature of prime numbers. Grounded in fundamental Set theory and Analysis, the proof uses the method of contradiction to demonstrate the absurdity of assuming the set of all prime numbers to be finite.

Riemann’s functional equation is valid on the vertical line of s=1/2. Each side is a real-valued function. The Riemann’s Xi function is also a real-valued function along the vertical line of s=1/2. Through the holomorphic extensions of the Riemann zeta function, starting from the real-valued function at s=1/2 into both sides of sigma<1/2 and sigma>1/2, we can get two versions of the zeta functional equation, eq. (45). The key property of the scaling and rotational factors g(s) and g(1-s) behave as multiplicative inverses in the complex plane, eq. (48). It is deduced that the Zeta function also has multiplicative inverses, the symmetric point is at (1/2,0) in the complex plane. The moduli behave as a hyperbola. Especially, along the vertical line sigam=1/2, the amplitudes of both function g(s) and g(1-s) are equal to 1, its arguments have opposite signs. If sigma is not equal to 1/2, the amplitudes of zeta(s) and zeta(1-s) are not equal to each other, because of their multiplicative inversion relationship. It is deduced that the non-trivial zeros can only be on the vertical line of s=1/2. A gamma function vector field is given in Appendix B, and some moduli of gamma function at special points are given. Finally, another variation of the Zeta function is provided in an integral form in Appendix D. The asymptotes behave as a c₈ cyclic group for the large t values.

In this research a neccessary and sufficient condition for the proof of the Binary Goldbach conjecture is established. It is established that the square of all natural numbers greater or equal to 2 have an additive partition equal to the sum of the square of a natural number greater or equal to zero and a Goldbach partition semiprime. All Goldbach partition semiprimes are odd except 4. This finding is in itself proof all composite even numbers have at least Goldbach partition. The result of the proof of the Binary Goldbach conjecture is used to prove the Andrica and Legendre conjectures. The Riemann hypothesis is examined and sources of non trivial zeroes outside the critical strip are discussed. An example example of a non-trivial zero outside the critical strip is given