Abstract: This paper presents a novel idea to compute the product of infinite geometric series on all prime numbers, which is equal to a series of natural numbers, that is, the sum of natural numbers up to infinite terms.

Abstract: 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.

Abstract: Abstract We study the moments of the function that counts the number of representations of an integer as sums of two prime squares. We refine some of the previous arguments and apply the Selberg sieve to get an unconditional upper bound for all moments. We also prove a lower bound for all moments conditional on some generalization of the Green-Tao theorem on linear equations in primes. More precisely, for the fifth moment and onward, we get the expected order of magnitude lower and upper bounds. In addition, we provide some heuristics on the mass function of this representation function.

Abstract: For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

Abstract: Abstract We address the prime geodesic theorem in arithmetic progressions and resolve conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the traces of closed geodesics on the modular surface do not equidistribute in the reduced residue classes of a given modulus.

Abstract: ABSTRACT. The Goldbach conjecture is one of the oldest unsolved problems in number theory and in the whole mathematical field. It is firstly mentioned by Prussian mathematician Christian Goldbach in his correspondence to Swiss mathematician Leonard Euler almost 300 years ago in 1742. The conjecture states that any even number greater than 2 could be build as sum of 2 prime numbers. Although there exist some proofs for heavily modified versions of the (odd) Goldbach conjecture, until this day, no proof exist for the original, 2-prime even Goldbach conjecture. In this paper after we have established some crucial statements around the conjecture and created a search algorithm to find the pairs of primes for any even numbers, we need to extend the Goldbach conjecture to the negative integer domain to be more able to use an important attribute of primes that they do exist in infinite numbers in both the positive and negative integer domain before we could proof the conjecture in its more generic and extended form using a probabilistic method. Keywords: Goldbach conjecture, prime numbers, sum, odd numbers, even numbers, extension, proof, negative integers, positive integers, probabilistic approach, even 2-prime form, generic form

Abstract: Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by R\'enyi. First we define the Midy property for a general real base $\beta >1$ and derive a necessary condition for validity of the Midy property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.

Abstract: We apply the Inclusion-Exclusion Principle to a unique pair of prime number subsequences to determine whether these subsequences form a small set or a large set and thus whether the infinite sum of the inverse of their terms converges or diverges. In this paper, we analyze the complementary prime number subsequences P(prime) and P(double-prime) as well as revisit the twin prime subsequence P2.

Abstract: The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod 8$. The idea of the proof in these cases is to find the class number for the quadratic imaginary field $\mathbb{Q}(i\sqrt p)$. Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is $p\equiv 7 \pmod 8$. We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete.

Abstract: The purpose of this work is to obtain exact and approximate formulas that calculate the number of partitions of even numbers into sums of pairs of prime numbers

Abstract: This study delves into the stochastic behavior of prime gaps, which are the differences between consecutive prime numbers, across various samples with sizes ranging up to 100 billion consecutive prime gaps. By analyzing multiple models approximating the probability mass function (PMF) of prime gaps, we evaluated their performance using statistical tests such as the Chi-Square Test, Kolmogorov-Smirnov Test, Mean Squared Error (MSE), and Kullback-Leibler Divergence. A key finding is the significance of incorporating the modular behavior of prime gaps with respect to 6 (g_n ≡ 0, 2, 4 (mod 6)) and adjusting the expectation of prime gaps in accurately predicting the approximated probability mass function (PMF) of prime gaps. Models that accounted for these modular properties and fine-tuned the expected value consistently outperformed those that did not. This research underscores the increasing complexity of expected prime gap behavior with larger n and suggests that prime gap distributions can be approximated by geometric distributions with parameter adjustments reflecting this complexity. The study’s findings have important implications for number theory and potential applications in cryptography. Future research directions include further exploration of modular properties, expanding the scope of statistical evaluations, and investigating additional mathematical properties influencing prime gap distributions.

Abstract: In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In this work we define a new notion: 'r-prime number of degree k' and we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1.

Abstract: : The X B number axis of a 0 to N 1 segment is shifted to the right by two number axis units and then overlapped in the same direction with the X A number axis of a 0 to N 1 segment. The overlapped number axis points of the two primes number are a set of twin primes and the law of twin primes smaller than N 1 is found. Then, the two X B number axes with the same length as them overlap in the opposite direction, and it is found that the overlapping number axis points of the two primes are the P (1,1) prime pairs of N 1 , which proves that N 1 can also be expressed as the sum of several groups of two primes. Comparing the two overlapping number axes, it is found that the X B \X B ′ overlapping number axes are symmetrical, and it is found that prime numbers that make 6P a exactly divide (N 1 +2P a ) or 6P b exactly divide (N 1 -2P b ) can also increase the table method number of P(1,1) prime pairs of N 1 , which proves that the number of P (1,1) primes of N 1 to the table group is about equal to or more than half of the number of twin primes smaller than N 1 .

Abstract: This paper addresses the computational methods and challenges associated with prime number generation, a critical component in encryption algorithms for ensuring data security. The generation of prime numbers efficiently is a critical challenge in various domains, including cryptography, number theory, and computer science. The quest to find more effective algorithms for prime number generation is driven by the increasing demand for secure communication and data storage and the need for efficient algorithms to solve complex mathematical problems. Our goal is to address this challenge by presenting two novel algorithms for generating prime numbers: one that generates primes up to a given limit and another that generates primes within a specified range. These innovative algorithms are founded on the formulas of odd-composed numbers, allowing them to achieve remarkable performance improvements compared to existing prime number generation algorithms. Our comprehensive experimental results reveal that our proposed algorithms outperform well-established prime number generation algorithms such as Miller-Rabin, Sieve of Atkin, Sieve of Eratosthenes, and Sieve of Sundaram regarding mean execution time. More notably, our algorithms exhibit the unique ability to provide prime numbers from range to range with a commendable performance. This substantial enhancement in performance and adaptability can significantly impact the effectiveness of various applications that depend on prime numbers, from cryptographic systems to distributed computing. By providing an efficient and flexible method for generating prime numbers, our proposed algorithms can develop more secure and reliable communication systems, enable faster computations in number theory, and support advanced computer science and mathematics research.

Abstract: In this work we have studied the prime numbers in the model P=am+1,m,a>1∈N. and the number in the form q=mam+bm+1in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cma=mam+1 is a generalized Cullen number then mam≡−1amodCma. In a second paper published in 2014, they also presented a test for Broth's numbers in Form kpn+1 where k1∈Nand p=qa+1where qis primeoddare special cases of the number mam+bm+1when btakes a specific value. For example, we proved if p=qa+1where q is odd prime and a>1∈N where πj=1qqjthen ∑j=1q−2πj−Cmaq−j−1q−am≡χmq−ammodp Components of proof Binomial theorem Fermat's Litter Theorem Elementary algebra.

Abstract: This paper introduces a novel prime-generating function f(p), which has been discovered to yield prime numbers for each prime input p. The function's properties and behavior are systematically analyzed, starting with the smallest prime, p=2, and iteratively applying the function with each subsequent prime output serving as the input for the next iteration. Through this process, a conjecture is proposed and rigorously proven, demonstrating that the function reliably generates prime numbers for all prime inputs. This discovery holds significant implications for number theory, offering new insights into the distribution and behavior of prime numbers. Furthermore, the discovery of this prime-generating function underscores the potential for further exploration and application in mathematical research.

Abstract: In April 2020, I proposed a conjecture regarding the distribution of prime numbers within specific intervals defined by square and oblong numbers. This conjecture was first discussed in the comments on sequences (https://oeis.org/A307508) and (https://oeis.org/A334163) on the Online Encyclopedia of Integer Sequences (OEIS). Sequence (https://oeis.org/A307508) identifies the primes located between a square number and its following oblong number, while sequence (https://oeis.org/A334163) lists the primes situated between an oblong number and its following square number.Building on this foundation, the present study proposes the following extensions of Legendre's Conjecture (inspired by discussions on https://www.mersenneforum.org/node/21664 and https://www.mersenneforum.org/node/21573):•There are at least 3 prime numbers between any two consecutive odd squares. Starting from 9 (the first odd composite square), there are at least 5 prime numbers between any two consecutive odd squares.•Including the prime odd square-minus-1 number 3, there are at least 5 prime numbers between any two consecutive odd square-minus-1 numbers.•Including the even prime number 2, there are at least 5 prime numbers between any two consecutive 0212-Oblong numbers.•There are at least 5 prime numbers between any two consecutive 0620-Oblong numbers.•Each of the four zones (0212-SO, 0212-OS, 0620-SO, 0620-OS) contains at least one prime number along the number line. Furthermore, starting from the odd square-minus-1 number 15, the 0620-SO zone contains at least two prime numbers along the number line.

Abstract: In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number Theorem for Arithmetic Progressions: If $k$ and $\ell$ are positive integers, with $1\le\ell\le k$ and $(\ell, k)=1$, then $$ \sum_{n\ge 2,\, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)}{n}=\frac{-1}{\phi(k)}, $$ where $\mu(n)$ is the Moebius function, $p(n)$ is the smallest prime factor of $n$, and $\phi(k)$ is the Euler function. Here we utilize the next level Duality identity between the second largest prime factor and the smallest prime factor, involving the Moebius function and $\omega(n)$, the number of distinct prime factors of $n$, to establish the following result as a consequence of the Prime Number Theorem for Arithmetic Progressions: For all $\ell$ and $k$ as above, $$ \sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)\omega(n)}{n}=0. $$ A quantitative version of this result is proved.

Abstract: In this paper, the distribution of prime numbers is expressed based on proving the Riemann hypothesis. The modality to the distribution of prime numbers is one of the most important results of 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 0.5. 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.

Abstract: 1. Based on the number theory of 0, 1, 0 + 1 = 1, 1 + 1 = 2, this paper constructs a positive integernatural number sequence.2. Any positive integer natural number N is composed of 1 (0 + 1 = 1 structural formula) or 2 (1 +1 = 2 structural formula) or 3 (2 + 1 = 3 or 1 + 1 + l = 3 structural formula) prime numbers, and iscomposed of 1 × P or 2 × P 2 .P 3 ..Pn or 3 (or a prime greater than 3) × P 3 . P 4 . The product of Pnprime numbers.3, 1 is the most basic and smallest prime number in elementary number theory. In modernmathematics, we all agree that it is not a prime number, and we all abide by it. But we also thinkthat it does not mean that it does not have the basic properties of prime numbers.4. Goldbach's conjecture is tenable; it is completely tenable when it is greater than or equal to 4(i.e 1 + 1 = 2).5. Explain the fundamental theorem of arithmetic, why there are only declarative expressionsand proof by reduction to absurdity in history.

Abstract: Prime numbers have long challenged mathematicians due to their seemingly random distribution along the number line. While equations easily capture the relationship between even and odd numbers, prime numbers present a unique challenge with no clear connection. In this paper, we delve into unraveling the hidden relationship among prime numbers, proposing a novelnotation to describe a simple yet accurate infinite series that encompasses all prime numbers on the number line. The introduction provides insights into the motivation behind this exploration.

Abstract: The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the set Q ′ ′ m = η / s : η ∈ Q ∗ m , s = 1 , 2 under the action of group H = < x ′ , y ′ : x ′ 2 = y ′ 4 = 1 > , where m is square free integer and Q ∗ m = a ′ + m / c ′ , a ′ , a ′ 2 − m / c ′ c ′ = 1 , c ′ ≠ 0 . We discuss the type and reduced cardinality of the orbit Q ′ ′ p . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime p . Further, we classify that for a reduced number r whether − r , r ¯ , − r ¯ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.

Abstract: 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: The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse χ 2 distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.

Abstract: &lt;p&gt;&lt;strong&gt;Abstract:&lt;/strong&gt; The Map of Integers and Divisors (MID) serves as a compelling visual paradigm, illustrating the complex interplay between integers and their divisors through a two-factor factorization landscape. This study extends our previous exploration of the MID's parabolic patterns, delving deeper into the arithmetic and geometric nuances manifested within the framework. We introduce the concept of SOM Zones&mdash;sequential zones demarcated by square, oblong, and (square minus 1) numbers&mdash;whose sizes increase linearly and pervade the numerical line.&lt;/p&gt; &lt;p&gt;Through meticulous analysis, we delineate the symbiotic relationship between divisors and these zones, using these connections to elucidate the prime number distribution along the numerical continuum. We investigate the reasons for the consistent emergence of at least one prime number between a square number and its consecutive oblong and another distinct prime between an oblong number and the subsequent square. These insights have led to the discovery of several novel prime number sieves&mdash;unprecedented in the literature&mdash;including SPM6: A Geometric Sieve of Prime Numbers in the MID Based on Multiples of 6, The MID Circles Sieve of Primes (MCSP), The MID Squares Sieve of Primes (MSSP), The MID Integers plus Divisors Sieve of Primes (MIPSP), and The MID Prime Integers minus Divisors Sieve (MIMSP).&lt;/p&gt; &lt;p&gt;These sieves lay the groundwork for a forthcoming study that expands upon these concepts, exploring their implications for the distribution of prime numbers within any polynomial framework. We conclude this preprint with a proposition for a "Factory for Record-Breaking Astronomical Prime Numbers," which leverages oblong numbers and large primorials in a novel methodological approach, setting the stage for potential breakthroughs in prime number discovery.&lt;/p&gt; &lt;p&gt;Please note that this document is a preprint and has not yet been peer-reviewed. It is intended for discussion and feedback within the scientific community to refine the approaches and conclusions presented.&lt;/p&gt; &lt;p&gt;&lt;strong&gt;Keywords: &lt;/strong&gt;Integers and Divisors, Pairs of Complementary Divisors, Factorization, Quadratic Polynomials, Prime Number Distribution, Sieve Methods in Number Theory, Parabolic Patterns, SOM Zones, Prime Sieves, Oblong Numbers, Primorials.&lt;/p&gt; &lt;p&gt;&lt;strong&gt;2020 Mathematics Subject Classification: &lt;/strong&gt;11A05 (Factorization of Integers), 11A41 (Primes), 11A51 (Factorization; primality), 11D09 (Quadratic and bilinear equations), 11N05 (Distribution of primes), 11N35 (Sieves), 11Y11 (Prime numbers: theory, sequences, sieves), 11Y55 (Calculation of integer sequences).&lt;/p&gt;

Abstract: This paper proposes a hypothesis regarding the distribution of prime numbers. The hypothesis is based on an expression and it suggests a consistent average number of primes in each group as the range of N increases. The distribution of prime numbers has been a subject of great interest in number theory. This paper presents a hypothesis that explores a specific expression involving the natural logarithm and the mathematical constants e and π. For N = 2 to 100, 000, 000:• Total Number of Groups: 415,236• Average Number of Primes in Groups: 15.00003101921015..