Prime Conjecture Research Articles

A Logical Proof of the Polignac's Conjecture Based on Partitions of an Even Number of a New Formulation

Polignac's Conjecture, proposed by Alphonse de Polignac in the 19th century, is a captivating hypothesis that extends the notion of twin primes to a broader context. It posits that for any even positive integer 2k, there exist infinitely many pairs of consecutive prime numbers whose difference is 2k. This conjecture is a natural generalization of the Twin Prime Conjecture, which focuses solely on pairs of primes differing by two. The conjecture has significant implications for our understanding of the distribution of prime numbers and the nature of their gaps and its exploration serves as a testament to the enduring fascination and mystery surrounding prime numbers and their properties. However, despite extensive efforts by mathematicians over the years, Polignac's Conjecture remains unproven, standing as one of the many unsolved problems in number theory. This study utilizes a set of all odd partitions generated from an even number of a new formulation, and we show that from this set of all pairs of odd numbers, there exist proper subsets containing infinitely many pairs of prime numbers whose differences is a fixed even gap. Finally, using these results and the facts that the difference of any two prime numbers is even and there exist infinitely many prime numbers, a logical proof of the Polignac's Conjecture is provided.

The Twin Prime Conjecture: An Analytical Approach

This paper examines the twin prime conjecture. The basic strategy is to first establish that there is no highest prime number by calculating the rates at which the multiples of each successive prime preclude higher numbers from being prime, and then proving that this rate (in the aggregate) can never reach 100%. The same basic methodology is then used to show that there can also be no highest twin prime.

We Can Differentiate Numbers, Too

Summary We survey the history of the arithmetic derivative and more recent advances in research on this topic. Among other things, we discuss a few generalizations of the original arithmetic derivative and some arithmetic differential equations that are related to Goldbach’s conjecture and the twin prime conjecture. Our primary purpose is to give an overview of this field, but we also aim at providing supplementary material for an introductory course on discrete mathematics or number theory. Therefore, our survey contains ten exercises.

Addendum to “On the orders of largest groups of automorphisms of compact Riemann surfaces” [J. Pure Appl. Algebra 225 (12) (2021) 106758

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . In the earlier paper we have showed together with Czesław Bagiński that there exist a rational sequence ( r n ) convergent to 8 and infinite sets R n so that μ ( g ) = r n ( g − 1 ) for g ∈ R n . Moreover, assuming the Twin Prime conjecture we have showed the same for 12 i.e. there exist a rational sequence ( s n ) convergent to 12 and infinite sets S n so that μ ( g ) = s n ( g − 1 ) for g ∈ S n . Here we prove the second assertion using Chen theorem instead of relying on TPC.

Advances in additive number theory

We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \pi(x) for enough large real x, being \pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \pi(x).

RETRACTED ARTICLE: The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I)

AbstractThe paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [4, 5]. The central idea consists in constructing an appropriate countable model $$\textbf{A}$$ A of Peano arithmetic by means of a Rasiowa–Sikorski set. This model validates the twin prime conjecture. Since $$\textbf{A}$$ A is elementarily equivalent to the standard model, the conjecture follows. Thus the standard model validates the twin primes conjecture. More generally, it is shown that de Polignac’s conjecture has a positive solution. The paper employs methods borrowed from the contemporary mathematical logic. Such a ’logical’ approach may be viewed as a useful addition to the dominant methodology in number theory based on mathematical analysis.

On the Chowla and twin primes conjectures over $\mathbb{F}_q[T

Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb{F}_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.

A Proof of the Twin Prime Conjecture in the Ƥ x Ƥ Space

A Proof of the Twin Prime Conjecture in the Ƥ x Ƥ Space

2-Odd Labeling of Graphs Using Certain Number Theoretic Concepts and Graph Operations

Graph theory plays a significant role in a variety of real-world systems. Graph concepts such as labeling and coloring are used to depict a variety of processes and relationships in material, social, biological, physical, and information systems. Specifically, graph labeling is used in communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd labeling assigns distinct integers to the nodes of <img src=image/13427595_01.png> in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, <img src=image/13427595_02.png>, <img src=image/13427595_03.png>. So, <img src=image/13427595_04.png> is a 2-odd graph if and only if it permits 2-odd labeling. Studying certain important modifications through various graph operations on a given graph is interesting and challenging. These operations mainly modify the underlying graph's structure, so understanding the complex operations that can be done over a graph or a set of graphs is inevitable. The motivation behind the development of this article is to apply the concept of 2-odd labeling on graphs generated by using various graph operations. Further, certain results on 2-odd labeling are also derived using some well-known number theoretic concepts such as the Twin prime conjecture and Goldbach's conjecture, besides recalling a few interesting applications of graph labeling and graph coloring.

Correlations of almost primes

AbstractWe prove that analogues of the Hardy–Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$ , under a restriction on the size of one of the prime factors of n and $n+h$ . Additionally, we consider correlations $n,n+h$ where n is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for almost all $|h| \leq H$ with $X^{1/6+\varepsilon}\leq H\leq X^{1-\varepsilon}$ .

Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers

A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: MN = 2N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0; MN]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

A New Method to Study Goldbach Conjecture

This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence); And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any odd prime sequence (3, 5, 7, …, Pn), there must have no LiKe sequence when the terms must be less than 3 × Pn. This method only studies prime numbers and corresponding composite numbers, replaced the relationship between even numbers and indeterminate prime numbers. In order to illustrate the importance of the idea of transforming the addition problem into the multiplication problem, we take the twin prime conjecture as an example and know there must exist twin primes in the interval [3Pn, P2n]. This idea is very important for the study of Goldbach conjecture and twin prime conjecture. It’s worth further study.

Infinitely Many Twin Prime Polynomials of Odd Degree

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over that differ by a fixed constant, for each . Elementary, constructive proofs were given for different cases by Hall and Pollack. In the same spirit, we discuss the construction of a further infinite family of twin prime tuples of odd degree, and its relations to the existence of certain Wieferich primes and to arithmetic properties of the combinatorial Bell numbers.

Numerical analysis approach to twin primes conjecture

The purpose of this paper is to demonstrate how the modified Sieve of Eratosthenes is used to have an approach to twin prime conjecture. If the Sieve is used in its basic form, it does not produce anything new. If it is used through the numerical analysis method explained in this paper, we obtain a specific counting of twin primes. This counting is based on the false assumption that distribution of primes follows punctually the Logarithmic Integral function denoted as Li(x) (see [5] and [10], pp. 174–176). It may be a starting point for future research based on this numerical analysis method technique that can be used in other mathematical branches.

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.

On the orders of largest groups of automorphisms of compact Riemann surfaces

Let μ ( g ) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g ≥ 2 . The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that μ ( g ) ranges between 8 ( g + 1 ) and 84 ( g − 1 ) and these two bounds are attained for infinitely many g . Here we prove that given a prime q > 167 congruent to −1 modulo 3, not congruent +1 modulo 5 and a prime p ≥ 84 q , μ ( g ) = ( 8 ( q + 4 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many integers m . Furthermore we also prove that for any prime q for which p = q + 2 is a prime, μ ( g ) = ( 12 ( q + 2 ) / q ) ( g − 1 ) for g = q p m + 1 for infinitely many m . These mean that there exist an infinite rational sequence ( r n ) convergent to 8 and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence ( s n ) convergent to 12 together with infinite sets R n and S n of integers for which μ ( g ) = { r n ( g − 1 ) if g ∈ R n s n ( g − 1 ) if g ∈ S n and 8 , 12 are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of μ . Namely if instead of μ = μ ( g ) we consider its normalization μ ˜ ( g ) = μ ( g ) / g then { 8 } ⊆ ( M d ) d ⊆ { 8 , 12 } , where M stands for the set of values of μ ˜ and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, 12 ∈ ( M d ) d .

A New Postulate and Some Conjectures Concerning Pair Primes in the Interval [n!, (n + k)!

This paper rather studies the behaviour of prime numbers bounded below and above by positive integers n! and (n + k)!, and then after some numerical evidence, postulates that there is at least one pair primes of gap k ∈ 2Z+ in between n! and (n + k)! for every integer n ≥ 2 and every even integer k &gt; 0. This assertion would eventually provide another structural form for Euclid theorem of ifinitude of primes, a kind of projection of the form in the original Bertrand postulate (now Chebychev's theorem). The truth- fulness of the conjecture that emanated from this postulate implies the Polignac's conjecture which aptly generalizes the twin prime conjecture. We thus present the new postulate and the conjectures for future research.

Hamiltonicity in Prime Sum Graphs

For any positive integer n, we define the prime sum graph $$G_n=(V,E)$$ of order n with the vertex set $$V=\{1,2,\cdots , n\}$$ and $$E=\{ij: i+j \text{ is } \text{ prime }\}$$ . Filz in 1982 posed a conjecture that $$G_{2n}$$ is Hamiltonian for any $$n\ge 2$$ , i.e., the set of integers $$\{1,2,\cdots , 2n\}$$ can be represented as a cyclic rearrangement so that the sum of any two adjacent integers is a prime number. With a fundamental result in graph theory and a recent breakthrough on the twin prime conjecture, we prove that Filz’s conjecture is true for infinitely many cases.

Distribution of adjacent prime numbers in the Ulam spiral

Purpose The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption. Design/methodology/approach This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes. Findings Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing. Originality/value A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.

There are Infinitely Many Fibonacci Primes

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.