Abstract: <p>This paper introduces a novel approach to estimating the distribution of of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. Application of this methodology to infinite sums and the nth sum, and propose several ways of defining the nth sum of a prime number By using the Ramanujan infinite series of natural numbers. I will be able to derive an infinite series of prime numbers.</p>

Abstract: We decompose natural numbers from structure which prime numbers have, as its starting point. With the decomposition, we can find a general law by categorization, which is in a power set and also in structure which prime numbers have, and we know that it limits the framework of structure about product and sum of natural numbers. In other words, $\sum_{k=1}^{n} \phi (k) \times [\frac{n}{k}] = \frac{n(n+1)}{2}$ holds, and it is equivalent to a basic formula of sum of divisors $\sum_{k|n} \phi (k) = n$.

Abstract: <p>Amazing.....? Negative and positive sum of prime number are equal...very interesting ...please see this working so beautiful. New conjectures in number theory I think</p>

Abstract: There are many theories and trials exist to find a structure for identification and prediction of prime numbers. The famous one is the Riemann Hypothesis. Beside this, there are running competitions to identify the prime numbers for enormous range of numbers with high performance computers. The idea to identify a structure in prime numbers is developed from an interview of Dr.Peter Plichta where he spoke on his book “Das Primzahlenkreuz”. Following is given a scheme that uses steps of separation to generate an order, which allows simple elimination steps to identify and predict prime numbers.

Abstract: Let q be a prime number and f(x)=xqs−axm−b be a monic irreducible polynomial of degree qs having integer coefficients. Let K=ℚ(𝜃) be an algebraic number field with 𝜃 a root of f(x). We give some explicit conditions involving only a,b,m,q,s for which K is not monogenic. As an application, we show that if p is a prime number of the form 32k+1, k∈ℤ and 𝜃 is a root of a monic polynomial f(x)=x2s−32cpx2r−p∈ℤ[x] with s>4,2∤c,s≠5+r, then ℚ(𝜃) is not monogenic.

Abstract: The distribution of primes over the set of natural numbers is a fascinating subject closely related to topics that range from the fundamental problem of factorization to applications in cryptography. Despite numerous efforts, efficient methods to locate huge primes are still under investigation. Here, we present an alternative approach to identifying prime numbers that is based on the evolution of the linear entanglement entropy. Specifically, we show that a singular behavior in the amplitudes of the Fourier series of this entropy is associated with prime numbers. We also discuss how this intriguing connection between primes and entanglement could be experimentally implemented using existing optical devices, and examine a possible relationship between our results and the zeros of the Riemann zeta function.

Abstract: <p>In this paper, we challenge the conventional wisdom surrounding prime numbers and their infinite nature, diverging from the Riemann hypothesis. We argue that prime numbers are, in fact, finite entities with profound implications for number theory. Our investigation reveals inherent flaws in number theory’s applicability to the real number line, which comprises an infinite continuum between whole numbers. We explore the connection to modular mathematics, highlighting the topological constraints on number division, ultimately leading to the neglect of the historical significance of whole numbers. Furthermore, we propose a novel perspective on the relationship between prime numbers and the fundamental principles of physics, particularly in the context of general relativity calculus. We assert that spacetime itself may be composed of prime number multiples of particle pairs, providing a solution to the potential consequences of forces acting with an infinite finitude through self-division. This paper challenges conventional mathematical and physical paradigms, offering a fresh perspective on the finite nature of prime numbers and its far-reaching implications for our understanding of the universe.</p>

Abstract: The Diophantine equation p^x + (p + 2q)^y = z^2 , where p, q and p + 2q are prime numbers, is studied widely. Many authors give q as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers p and q for showing that the Diophantine equation p^x + (p + 2q)^y = z^2 has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

Abstract: Abstract We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes $$p-1$$ p - 1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

Abstract: In Chapter 2, we made extensive use of the fact that every positive integer can be written in one and only one way as a product of powers of distinct primes. This property of $$\mathbb Z$$ is basic to mathematics. It is so basic that many people don’t even think to question it. This is especially true in school, where students spend much of elementary school working with integers, using this unique factorizationUnique factorization property as if it were a law of nature. For example, young children build “factor trees” for whole numbers, and it is usually taken for granted that two different trees, like those in Figure 3.1, end up with the same set of prime factors.

Abstract: The multiparty comparison allows to compare two integers $x$ and $y$ blindly, where a set of players hold the shares of the elements $x and y, x, y \in \mathbb {F}_{p}$ , a prime field. The existing multiparty comparison protocols execute in constant rounds, but the number of multiplications depends on the size of the prime $p$ , i.e., the communication complexity will be high for large prime $p$ . In this paper, we present a multiparty comparison protocol with constant rounds in which the number of multiplications depends on the number of players rather than the prime $p$ itself. This multiparty comparison protocol is further extended to design a multiparty equality-test protocol. An equality-test protocol computes the equality of shares in constant rounds and its number of multiplications depends on the number of players. Our proposed protocols multiparty comparison and equality-test are unconditionally secure against the active and passive attacks and have $O(n)$ communication complexity, where $n$ is the number of players. We also present an efficient technique for fault detection that can verify the correctness of various protocols.

Abstract: <p>In the pursuit of understanding the enigmatic world of prime numbers, a unique formula has been identified, which can be used to generate prime numbers, with the exception of 2 and 3. This formula is expressed as (n = positive integer, p = prime number), where, intriguingly, the output for other positive integers results in irrational numbers. This enigmatic formula not only reveals prime numbers but also unveils a peculiar pattern related to composite numbers derived from prime factors. These composite numbers, which arise as exceptions in the context of prime generation, exhibit regularity and may offer new insights into the interconnected of prime and composite numbers. This discovery promises to broaden our understanding of the intricate world of number theory and provides a fresh perspective on the nature of prime numbers. Further exploration of this formula may uncover deeper mathematical principles and unlock novel avenues in number theory research.</p> <p>p =√1 + 24n</p>

Abstract: <p> Abstract. In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture. </p>

Abstract: <p>The “strong Goldbach conjecture” posits that any even number exceeding 6 can be represented as the sum of two prime numbers. This study explores this hypothesis, leveraging the constancy of odd integer quantities and cumulative sums within positive integers. By identifying odd prime numbers, pα1 and pα2, within [3, n] and (n, 2n-2) intervals, we demonstrate a transformative process grounded in the unchanging nature of odd number counts and their cumulative sums. Through this process, we establish the equation 2n =pα1 + pα2, offering a significant stride in unraveling the enigmatic core of the strong Goldbach conjecture.</p>

Abstract: The efficiency with which an integer may be factored into its prime factors determines several public key cryptosystems&prime; security in use today. Although there is a quantum-based technique with a polynomial time for integer factoring, on a traditional computer, there is no polynomial time algorithm. We investigate how to enhance the wheel factoring technique in this paper. Current wheel factorization algorithms rely on a very restricted set of prime integers as a base. In this study, we intend to adapt this notion to rely on a greater number of prime integers, resulting in a considerable improvement in the execution time. The experiments on composite numbers n reveal that the proposed algorithm improves on the existing wheel factoring algorithm by about 75%

Abstract: Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.

Abstract: Abstract We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular, we prove for $$ \begin{align*} & \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \end{align*}$$an asymptotic formula that holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$, we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

Abstract: 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 &ldquo;Ueber die Anzahl der Primzahlen unter einer gegebenen Gr&ouml;sse&rdquo; (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 &zeta;(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 &zeta;(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 &zeta;(z)=0 Proof that |X(z)|=|Y(z)| only if Re(z)=1/2 Conclude that &zeta;(z)=0 only if Re(z)=1/2 for Re(z) 0 PART 5: We will also prove that all non-trivial zeros of &zeta;(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.

Abstract: A BSTRACT . In mathematics, the prime counting function π ( x ) is defined as the function yielding the number of primes less than or equal to a given number x . In this paper, we prove that the asymptotic limit of a summation operation performed on a unique subsequence of the prime numbers yields the prime number counting function π ( x ) as x approaches ∞ . We also show that the prime number count π ( n ) can be estimated with a notable degree of accuracy by performing the summation operation on the subsquence up to a limit n .

Abstract: Mathematics is a subject that has been developed for thousands of years. Alongside everything else, prime numbers and perfect numbers are studied since the earliest of time, and still today. Despite all the improvements in mathematics and computer power, mysteries of prime numbers and perfect numbers still remain. From trial division, to Mersenne primes and Fermat numbers, all the way to Reimann Hypothesis, mathematicians kept attempting to capture the pattern of primes. Although a lot of progress are made, it is still not known precisely. Searching of big primes and constructing special composite numbers contributed to other areas a lot, from encryption to proofs of other number theoretic concepts. Perfect numbers are even rarer than primes. The study of perfect numbers can be traced back to the Pythagorean brotherhood. In this article, we will discuss about brief history of the research, theorems, proofs, as well as some useful algorithms.

Abstract: The efficiency with which an integer may be factored into its prime factors determines several public key cryptosystems’ security in use today. Although there is a quantum-based technique with a polynomial time for integer factoring, on a traditional computer, there is no polynomial time algorithm. We investigate how to enhance the wheel factoring technique in this paper. Current wheel factorization algorithms rely on a very restricted set of prime integers as a base. In this study, we intend to adapt this notion to rely on a greater number of prime integers, resulting in a considerable improvement in the execution time. The experiments on composite numbers n reveal that the proposed algorithm improves on the existing wheel factoring algorithm by about 75%.

Abstract: In this paper, we suggest a new approach to find any prime numbers up to a given n ∈ ℕ*. The proposed procedure does not work like a sieve and is easy to implement as it only uses assignments and subtractions which lead to improvements in memory requirements and upgradeable runtime performance. This is because, also, the algorithm suits well parallel computing. These results aim to solve several problems affecting those routines based on sieve methods, especially when large numbers are considered.

Abstract: We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.

Abstract: <p> Abstract. In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture. </p>

Abstract: Abstract Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet L-functions is true, we then establish explicit formulae for $\psi(x,\chi)$, $\theta(x,\chi)$ and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.

Abstract: We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.

Abstract: Lucas sequence is one of the most studied binary recurrence sequence defined by the relation $L_{n+2}=L_{n+1}+L_n;~L_0=2, L_1=1$. In this paper, we investigate all the sums and differences of two Lucas numbers that are powers of a odd prime $p$ satisfying $p<10^3$.