5,961 Results Found

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes fol...

The arithmetic average of the first n primes, ${\overline{p}}_n={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{p}_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n\to {\overline{p}}_n$ is extremely easy to invert, $p_n=n{\overline{p}}_n-(n-1){\overline{p}}_{n-1}$, it...

Let $q>r\ge 1$ be coprime integers. Let ${\mathbb{P}}_c={\mathbb{P}}_c(q,r,\mathcal{H})$ be an increasing sequence of primes p satisfying two conditions: (i) $p\equiv r$ (mod q) and (ii) p starts a prime k-tuple with a given pattern $\mathcal{H}$. L...

In this article, we study some new properties of twin primes and algorithms for their generation. We find the necessary conditions to generate a pair of twins. These conditions seem to indicate that the conjecture is true, namely, there are infinitel...

The security of RSA relies on the computationally challenging factorization of RSA modulus $N=p_1 p_2$ with $N$being a large semi-prime consisting of two primes $p_1\mathrm{and} p_2,$ for the generation of RSA keys in commonly adopted cryptosystems. The...

One of the themes of this paper is recent results on large gaps between primes. The first of these results was achieved in the paper by Ford, Green, Konyagin and Tao. It was later improved in the joint paper of these four authors with Maynard. One of...

The Lambert W function, implicitly defined by $W\left(x\right){}^{}$...

For several centuries, prime factorisation of large numbers has drawn much attention due its practical applications and the associated challenges. In computing applications, encryption algorithms such as the Rivest–Shamir–Adleman (RSA) cr...

In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by $n=pr$ with prime number p. We a...

The invention in 1978 of the first practical asymmetric cryptosystem known as RSA was a breakthrough within the long history of secret communications. Since its inception, the RSA cryptosystem has become embedded in millions of digital applications w...

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers...

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates thro...

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space $(\mathfrak{LS},$ ${\tau }^0)$ induced by measurable functions on p-adic number fields ${\mathbb{Q}}_p$ over primes $p$ . In pa...

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the a...

A conjecture of Cai–Zhang–Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper, we give an equivalent proposition to the conjecture. By considering extreme value problems with constraint...

Let $\alpha ,\beta \in \mathbb{R}$ with $\alpha \ne 0$, and let $\gamma \in (0,5/6)$. Define the set $M_1$ to consist of primes p such that $p+2$ is almost prime, and let $M_2$ be the set of primes of the form $p=a^2+b^2+1$. We study the distribution of $\alpha p^{\gamma }\text{\&nbs}$...

Let ${\lambda }_1,{\lambda }_2,{\lambda }_3$ be nonzero real numbers, not all of the same sign; let ${\lambda }_1/{\lambda }_2$ be irrational; and let $\eta$ be any real number. We investigate the solvability of the inequality $|{\lambda }_1{p}_1+{\lambda }_2{p}_2+{\lambda }_3{p}_3^2+\eta$...

Medical students (MSs) are healthcare workers and must also cope with education-related stressors. This study aims to assess factors associated with perceived stress in Italian MSs. A cross-sectional study was conducted in 12 Italian medical schools...

Several attacks on the well-known RSA cryptosystem that can be extended to a multi-prime version of RSA reveal that it is preferable to use the modulus having more prime factors. On the contrary, the larger the number of prime factors of the modulus,...

Let x be a large real number, $d\left(n\right)$ be the Dirichlet divisor function and $\Delta \left(x\right)={\sum }_{n⩽x}d\left(n\right)-xlogx-(2\gamma -1)x$. In this paper, we consider a weighted form of the three primes theorem: $S(N;k_1,k_2,k_3):={\sum }_{p_1+p_2+p_3=N}{\&Delt}^{}$...

Mersenne prime numbers, expressed in the form (2ⁿ − 1), have long captivated researchers due to their unique properties. The presented work aims to develop a symmetric cryptographic algorithm using a novel technique based on the logical propert...

By chunking continuous streams of action into ordered, discrete, and meaningful units, event segmentation facilitates motor learning. While expertise in the observed repertoire reduces the frequency of event borders, generalization of this effect to...

Tomato plants can establish symbiotic interactions with arbuscular mycorrhizal fungi (AMF) able to promote plant nutrition and prime systemic plant defenses against pathogens attack; the mechanism involved is known as mycorrhiza-induced resistance (M...

Managing soil-borne pathogens is complex due to the restriction of the most effective synthetic fungicides for soil treatment. In this study, we showed that seed priming with Jania adhaerens water-soluble polysaccharides (JA WSPs) was successful in p...

In this paper, we study certain Banach-space operators acting on the Banach *-probability space ( $\mathbb{LS}$ , ${\tau }^0$ ) generated by semicircular elements ${\Theta }_{p,j}$ induced by p-adic number fields ${\mathbb{Q}}_p$ over the set $\mathcal{P}$ of...

The ongoing global emergence of arthropod-borne (arbo) viruses has accelerated research into the interactions of these viruses with the immune systems of their vectors. Only limited information exists on how bunyaviruses, such as Rift Valley fever vi...

In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to...

Macrophages undergo different cellular states upon activation that can be hyporesponsive (tolerated) or hyperresponsive (primed or trained) to subsequent stimuli. Epigenetic modifications are known to play key roles in determining these cellular stat...

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expa...

Conditions that are necessary for the relative annihilator in lower $BCK$ -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime fact...

Antibiosis is a key feature widely exploited to develop biofungicides based on the ability of biological control agents (BCAs) to produce fungitoxic compounds. A less recognised attribute of plant-associated beneficial microorganisms is their ability...

Syntactic priming in dialogue occurs when exposure to a particular syntactic structure implicitly induces a speaker’s subsequent preference for the same syntactic structures in their own speech. Here, we asked whether this priming effect is boo...

Visual attention is guided by the history of selections in previous trials, an effect usually referred to as intertrial priming. The aim of the present study was to investigate whether such priming in visual search is due to a strengthening of the ta...

Consider both the Logarithmic integral, $\mathrm{Li}\left(x\right)={lim}_{ϵ\to 0}\left\{{\int }_0^{1-ϵ}{\displaystyle \frac{du}{lnu}}+{\int }_{1+ϵ}^x{\displaystyle \frac{du}{lnu}}\right\}$, and the prime counting function $\pi \left(x\right)={\sum }_{p\le x}1$. From several recently developed known effective bounds on the prime counting functio...

In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and gra...

Plant priming is an induced physiological state where plants are protected from biotic and abiotic stresses. Whether seaweed extracts promote priming is largely unknown as is the mechanism by which priming may occur. In this study, we examined the ef...

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ide...

In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an alge...

Biostimulants are the next-generation choice for sustainable agricultural production and are gradually becoming an alternative to synthetic chemicals. Various botanicals are proposed to exert stimulatory effects, and garlic allelochemicals are among...

Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the dis...

The systematic generation of prime numbers has been almost ignored since the 1990s, when most of the IT research resources related to prime numbers migrated to studies on the use of very large primes for cryptography, and little effort was made to fu...

Cadmium (Cd) is considered one of the most toxic heavy metals to living organisms, being very persistent in soil and non-biodegradable, thus posing a long-term hazard to plants and humans. In recent years, the application of different molecules at th...

To ameliorate plants’ response to environmental stresses, seed priming can be a useful tool; it consists of the pre-exposure of the seeds to mild stress, which improves plant adaptation to future exposure to adverse growth conditions. In our previous...

The influence of subliminal priming (behavior outside of awareness) in humans is an interesting phenomenon and its understanding is crucial as it can impact behavior, choices, and actions. Given this, research about the impact of priming continues to...

All rings considered are commutative with identity, and all modules are assumed to be unital. In this paper, we study R-modules in which every quasi-primary submodule is also primary; we refer to such modules as satisfying condition (*). We present s...

A self-priming centrifugal pump can be used in various areas such as agricultural irrigation, urban greening, and building water-supply. In order to simulate the gas-water two-phase flow in the self-priming process of a self-priming centrifugal pump,...

This manuscript addresses the phenomenon of masked priming and the cognitive process of switching from Spanish to English while reading in sequential bilingual texts compared to heritage speakers. A lexical decision task was employed in the present s...

Combating environmental stress related to the presence of toxic elements is one of the most important challenges in plant production. The majority of plant species suffer from developmental abnormalities caused by an exposure to toxic concentrations...

This paper identifies all derivations associated with two well-known non-commutative prime rings and provides some remarks on one of these derivations, called the prime derivation. Also, it presents two results for some classes of non-commutative pri...

This study investigates the cross-linguistic priming effect in the syntactic written output of late bilingual Levantine Arabic speakers who learn English as a second language. In particular, we examined priming sentence type (simple vs. complex sente...