Chaos in Physics and recurrence in Arithmetic Sets

Thedecades around 1800were not a period inwhich puremathematics in general, and number theory in particular, flourished in Italy, see [Bottazzini 1994]. It is significant in this respect that Joseph Louis Lagrange, whose birth and early studies took place in Torino, finally became a prominent representative of the Frenchmathematical school and that, decades later, Guglielmo Libri still spent most of his academic career in France. Thus, Gauss’s Disquisitiones Arithmeticae did not have an immediate resonance in Italian mathematical circles. Gianfrancesco Malfatti, a professor in Ferrara, already seventy years old at the time of the publication of theDisquisitiones Arithmeticae, was one of the rare Italian mathematicians who manifested some interest in arithmetic, and above all in the theory of algebraic equations, in the last decades of the eighteenth century. In 1804, he criticized Ruffini’s first attempts to prove that equations of degree higher than 4 are not solvable by radicals and thus contributed to Ruffini’s later works. A year later, Malfatti published what was probably the best Italian paper on arithmetical matters at that time, [Malfatti 1805], in which he studied the following problem using simple divisibility properties: multiply a given integer by a square, in such a way that it may be expressed as a sum of a given number of squares. However, none of these papers mentions Gauss and his masterwork. In fact, Malfatti did not display a serious knowledge of the recent developments in number theory; the short historical discussion in his 1805 paper only says:

This paper contains a summary table of the author’s computation of the Gaussian primes of the form a + i a + i . For the values x = 1000 , 10000 , 100000 , 180000 x = 1000, 10000, 100000, 180000 , and 500000 (500000) 14000000, the following values are tabulated: G ( x ) G(x) , the numbers of Gaussisn primes a + i a + i with a ≦ 14000000 ; π 1 ( x ) a \leqq 14000000;{\pi _1}(x) , the number of primes ≦ x \leqq x congruent to 1 m o d 4 1 mod 4 ; π 3 ( x ) {\pi _3}(x) , the number of primes ≦ x \leqq x congruent to 3 m o d 4 3 mod 4 ; and G ( x ) / π 3 ( x ) G(x)/{\pi _3}(x) .

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

With the availability of microcomputers, mathematics classes have the opportunity to explore topics in number theory that have previously been ignored or treated superficially. The exploration of problems in number theory permits a further understanding of the BASIC language by introducing new computer functions. One topic that couples number theory to programming is palindromes.

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …

Prime numbers are an important research area with application in computer security. The security of certain cryptographic schemes critical in information security lies in the difficulty of factoring large numbers into their prime factors. Determination of the distribution of prime numbers is an important unsolved problem in number theory. Biomolecules such as DNA have extended the domains of what is considered a standard computer model. This report investigated the distribution of prime numbers generated from viral DNA. The digits of Euler’s number and Pi were found to be encoded between Bacteriophage T4 Watson-Crick DNA segments comprised of prime numbers. Key words: Prime number, number theory, molecular computing, phage T4.

Cryptographic primitives have been constructed on a wide range of mathematical basis. For asymmetric cryptography, those bases are problems in number theory. Lattice based mathematical problems provide a basis for cryptographic algorithms. Lattice-based cryptographic algorithms are believed to be resistant to quantum computing attacks. This makes research concerning these algorithms important. What is lacking in the current literature is a comparative study of existing cryptanalysis data for these algorithms. That data can be used to guide further research into lattice-based asymmetric cryptographic primitives. This current paper provides such a comparison.

Cryptography is a hot and active topic, it plays a crucial role in many aspects nowadays, from internet banking and e-commerce to email and web-based business processes. An important area of research today is to test the security of cryptosystems. A cryptographic system is safe as long as no one could not break it. Modern cryptographic algorithms are based on mathematical problems, known to be difficult, so breaking the security of a cryptosystem requires the solution of a such problem, as in the case of RSA, which is based on the factorization problem in number theory. This paper proposes a symmetric-key cryptosystem named Rotating Disk Cryptosystem and is denoted as RDC used for encryption and decryption of text as well as images which is based on chaotic function and the factorization problem.

This paper systematically constructs a unified solution method for polynomial equations based on differential algebraic closure and proves its strict equivalence to classical transcendental function solutions.The core contributions lie in: (1) Constructing the universal differential algebraic closure Kn and its specialization homomorphism, providing explicit finite closed-form solutions for polynomials of any degree n; (2) Proving explicit isomorphisms between these differential algebraic solutions and solutions expressed via trigonometric, hyperbolic, elliptic, hyperelliptic, and Riemann θ functions, thereby building a bridge between finite operations (arithmetic, differentiation, evaluation, radicals) and transcendental functions;(3) Conversely demonstrating that a large class of transcendental functions (solutions of algebraic differential equations) can be finitely represented by differential algebraic solutions of polynomial equations, thus precisely characterizing the class of “finitely representable functions” and clarifying its relativity—dependent on the chosen set of basic operations and constants; (4) Based on this unified framework, proposing innovative, potentially computable differential algebraic research pathways for cutting-edge number theory problems (including the Birch–Swinnerton-Dyer (BSD) conjecture, class number problems, the abc conjecture, and an explicit formulation of the Langlands program). This work not only expands the boundaries of the Abel–Ruffini theorem by introducing and characterizing the differential algebraic complexity class DA, but also unifies the classical spectrum of polynomial equation solving—from quadratic, cubic, quartic to quintic and beyond—within an explicit, coefficient-driven finite representation framework, offering potential constructive proof tools for several outstanding problems in number theory and arithmetic geometry.

Dynamic Bayesian networks (DBNs) are compact graphical representations used to model probabilistic systems where interdependent random variables and their distributions evolve over time. In this paper, we study the verification of the evolution of conditional-independence (CI) propositions against temporal logic specifications. To this end, we consider two specification formalisms over CI propositions: linear temporal logic (LTL), and non-deterministic Büchi automata (NBAs). This problem has two variants. Stochastic CI properties take the given concrete probability distributions into account, while structural CI properties are viewed purely in terms of the graphical structure of the DBN. We show that deciding whether a stochastic CI proposition eventually holds is at least as hard as the Skolem problem for linear recurrence sequences, which is a long-standing open problem in number theory. On the other hand, we show that verifying the evolution of structural CI propositions against LTL and NBA specifications is in PSPACE, and is hard for both NP and coNP. We also identify natural restrictions on the graphical structure of the DBN that make the verification of structural CI properties tractable.

We complete the investigations in [11] on the classification of connected components of moduli of real polarized -surfaces. In particular, we show that this classification is closely related to some classical problems in number theory: the classification of binary indefinite lattices and the representation of integers as sums of two squares. As an application, we use recent results in [13] to completely classify real polarized -surfaces that are deformations of real hyperelliptically polarized -surfaces. This is important because real hyperelliptically polarized -surfaces can be constructed explicitly.

Let 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.

Participants at an October workshop at the Mathematical Sciences Research Institute in Berkeley, California, learned that a team of mathematicians has proved a deep similarity between a simple form of solitaire and a mathematical tool called random matrices, originally developed to understand the quantum behavior of large atoms. The implications could go well beyond card games to some of the most puzzling patterns in mathematics. Other recent work suggests that the same random matrix key might unlock the most important problem in number theory: proving the Riemann hypothesis, which describes how prime numbers are distributed among other integers.