Open Problem In Number Theory Research Articles

New explorations and remarkable inequalities related to Fortune’s conjecture and fortunate numbers

Fortune's conjecture (named after the social anthropologist Reo Franklin Fortune) is an extremely elegant mathematical conjecture that always remains an open problem in number theory. It is a conjecture about prime numbers, which leads to the so-called "fortunate numbers" (not to be confused with "lucky numbers"): Reo F. Fortune predicted that no fortunate number is composite. This conjecture impresses us all as mathematicians, that's why we decided that it will be the subject of this paper, which has many objectives and very interesting findings, among them: - Highlighting numerous properties of the fortunate numbers. - Giving a proof of Fortune's conjecture in a particular case, by two different methods one of which is original. - Presenting many counterexamples that reinforce the previous point, when the satisfied hypothesis in that particular case, is not met. - Proving a new remarkable inequality that all the 3000 first (known until now) fortunate numbers perfectly fulfill. Despite our continuous research (quite recently) on the subject of Fortune’s conjecture, we have never found a mathematical reference with a variety of ideas dealing with this conjecture, so we hope that this paper will be the first scientific work containing multiple ideas, comments, results and goals, and contributing significantly to find a definitive solution of Fortune's conjecture, therefore this paper may advance the field.

Erratum to: A common approach to three open problems in number theory

Erratum to: A common approach to three open problems in number theory

A common approach to three open problems in number theory

A common approach to three open problems in number theory

Subindices and subfactors of finite groups

Recently sub-factors, sub-indices and infinite difference of group subsets have been introduced as a research topic by the author. We assign four sub-indices and the -difference length to every subset A of a group G. We note that all sub-indices are equal (i.e., A is index stable in G) if A is a subgroup or a normal sub-semigroup of G. It is proved that index stability of primes in integers is equivalent to a well-known open problem in number theory. In order to develop the theory, in this paper, we study sub-index properties in finite groups and give some criteria for the index stability. Also, an algorithm for finding sub-factors of finite groups is obtained. As an application of the topic, we provide some lower bounds for the order of the subgroup generated by a given subset of finite groups. Finally, we state many questions, problems, conjectures and some future directions of the topic.

On elliptic curves with p-isogenies over quadratic fields

AbstractLet K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

A Prime Testing Algorithm from Leonhard Euler

In 1749, Leonhard Euler solved a longstanding open problem in number theory, proving Fermat’s 1640 conjecture that any prime number of the form can be written as the sum of two squares. This achievement overshadowed an equally impressive calculation that Euler devised at the same time, to test numbers for primality. In this article, we explore Euler’s primality test, demonstrating its computational parity with trial division, at least for a class of testable numbers. Additionally, we will see that primality testing was one of Euler’s lifelong interests, and was a topic he returned to time and again as an application of his number theory work.

Mathematical Modelling of COVID-19 and Solving Riemann Hypothesis, Polignac's and Twin Prime Conjectures Using Novel Fic-Fac Ratio With Manifestations of Chaos-Fractal Phenomena

COVID-19 originated from Wuhan, China in December 2019. Declared by the World Health Organization on March 11, 2020; COVID-19 pandemic has resulted in unprecedented negative global impacts on health and economy. International cooperation is required to combat this ”Incompletely Predictable” pandemic. With manifestations of Chaos-Fractal phenomena, we mathematically model COVID-19 and solve [unconnected] open problems in Number theory using our versatile Fic-Fac Ratio. Computed as Information-based complexity, our innovative Information-complexity conservation constitutes a unique all-purpose analytic tool associated with Mathematics for Incompletely Predictable problems. These problems are literally ”complex systems” containing well-defined Incompletely Predictable entities such as nontrivial zeros and two types of Gram points in Riemann zeta function (or its proxy Dirichlet eta function) together with prime and composite numbers from Sieve of Eratosthenes. Correct and complete mathematical arguments for first key step of converting this function into its continuous format version, and second key step of using our unique Dimension (2x -N) system instead of this Sieve result in primary spin-offs from first key step consisting of providing proof for Riemann hypothesis (and explaining closely related two types of Gram points), and second key step consisting of providing proofs for Polignac’s and Twin prime conjectures.

SICs: Some Explanations

The problem of constructing maximal equiangular tight frames or SICs was raised by Zauner in 1998. Four years ago it was realized that the problem is closely connected to a major open problem in number theory. We discuss why such a connection was perhaps to be expected, and give a simplified sketch of some developments that have taken place in the past 4 years. The aim, so far unfulfilled, is to prove existence of SICs in an infinite sequence of dimensions.

Chebotarev density theorem in short intervals for extensions of 𝔽_{𝕢}(𝕋)

An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E E of Q \mathbb {Q} with Galois group G G , a conjugacy class C C in G G , and a 1 ≥ ε > 0 1\geq \varepsilon >0 , one wants to compute the asymptotic of the number of primes x ≤ p ≤ x + x ε x\leq p\leq x+x^{\varepsilon } with Frobenius conjugacy class in E E equal to C C . The level of difficulty grows as ε \varepsilon becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1 / 2 1\geq \varepsilon >1/2 . We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0 \varepsilon >0 . Our result is valid in the limit when the size of the finite field tends to ∞ \infty and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G G -factorization arithmetic functions.

On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

In this paper, we give a survey of the known results concerning the tensor rank of multiplication in finite extensions of finite fields, enriched with some unpublished recent results, and we analyze these to enhance the qualitative understanding of the research area. In particular, we identify and clarify certain partially proved results and emphasise links with open problems in number theory, algebraic geometry, and coding theory. Bibliography: 92 titles.

Forcing the truth of a weak form of Schanuel’s conjecture

Schanuel’s conjecture states that the transcendence degree over ℚ of the 2n-tuple (λ 1 ,⋯,λ n ,e λ 1 ,⋯,e λ n ) is at least n for all λ 1 ,⋯,λ n ∈ℂ which are linearly independent over ℚ; if true it would settle a great number of elementary open problems in number theory, among which the transcendence of e over π.

GENERATION OF SECOND MAXIMAL SUBGROUPS AND THE EXISTENCE OF SPECIAL PRIMES

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^{r}-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

$abc$ triples

The $abc$ conjecture, one of the most famous open problems in number theory, claims that three relatively prime positive integers $a,b,c$ satisfying $a+b=c$ cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than $c$. Triples of relatively prime numbers satisfying $a+b=c$ are called {\em $abc$ triples} if the product of their distinct prime divisors is strictly less than $c$. We catalog what is known about $abc$ triples, both numerical examples found through computation and infinite familes of examples established theoretically. In addition, we collect motivations and heuristics supporting the $abc$ conjecture, as well as some of its refinements and generalizations, and we describe the state-of-the-art progress towards establishing the conjecture.

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition $$+$$ and multiplication $$\times $$ are replaced by the operations maximum $$\oplus $$ and addition $$\otimes $$ , in other words $$a\oplus b=\max \{a,b\}$$ and $$a\otimes b=a+b$$ .

Decidability of Approximate Skolem Problem and Applications to Logical Verification of Dynamical Properties of Markov Chains

When studying probabilistic dynamical systems, temporal logic has typically been used to analyze path properties. Recently, there has been some interest in analyzing the dynamical evolution of state probabilities of these systems. In this article, we show that verifying linear temporal properties concerning the state evolution induced by a Markov chain is equivalent to the decidability of the Skolem problem -- a long-standing open problem in Number Theory. However, from a practical point of view, usually it is enough to check properties up to some acceptable error bound ϵ. We show that an approximate version of the Skolem problem is decidable, and that it can be applied to verify, up to arbitrarily small ϵ, linear temporal properties of the state evolution induced by a Markov chain.

The Stickelberger splitting map and Euler systems in the K -theory of number fields

For a CM abelian extension F / K of a totally real number field K , we construct the Stickelberger splitting maps (in the sense of Banaszak, 1992 [1] ) for the étale and the Quillen K -theory of F and use these maps to construct Euler systems in the even Quillen K -theory of F . The Stickelberger splitting maps give an immediate proof of the annihilation by higher Stickelberger elements of the subgroups div K 2 n ( F ) l of divisible elements of K 2 n ( F ) ⊗ Z l , for all n > 0 and all odd primes l . This generalizes the results of Banaszak (1992) [1] , which only deals with CM abelian extensions of Q . Throughout, we work under the assumption that the Iwasawa μ -invariant conjecture holds. In upcoming work, we will use the Euler systems constructed in this paper to obtain information on the groups of divisible elements div K 2 n ( F ) l , for all n > 0 and odd l . The structure of these groups is intimately related to some long standing open problems in number theory, e.g. the Kummer–Vandiver and Iwasawa conjectures.

Discrete asymptotic deterministic randomness for the generation of pseudorandom bits

The deterministic randomness not only can become a dominant approach in exploring the relationship between chaos and randomness, but also can be associated with some famous number theoretical concepts and open problems in number theory. Compared with chaotic sequences, asymptotic deterministic randomness sequences have the characteristic of multi-value correspondence, which makes those sequences unpredictable in short steps. In this Letter, we will propose the definition of the discrete asymptotic deterministic randomness, and then analyze the dynamical characteristics such as maximum-period and multi-value correspondence. Referring to the NIST800-22 statistical test suite, we will present and discuss two examples of PRBGs based on DADR, from the point of view of FPGA design and randomness quality.

Some problems in number theory that arise from group theory

In this expository paper, we present several open problems in number theory that have arisen while doing research in group theory. These problems are on arithmetical functions or partitions. Solving some of these problems would allow to solve some open problem in group theory.

A Characterization of Refinable Rational Functions

In recent decades, refinable functions have become increasingly popular due to their desirable properties in many applications. Rational functions, however, are not as well-behaved as some other classes of functions and have seemingly escaped notice in terms of refinability. The authors spent the summer of 2006 investigating the refinability of rational functions while attending a National Science Foundation funded Research Experience for Undergraduates program at Texas A & M University. Preliminary simplifications to the general case are presented in a chronological collection of lemmas. A complete characterization of refinable rational functions follows with an interesting connection to an open problem in number theory.