METALLIC RATIOS AND THEIR SYMMETRY GROUPS: A DEEP ARITHMETIC-GEOMETRIC PERSPECTIVE

Computational number theory, a branch of mathematics focused on the computational aspects of number theory, has witnessed rapid advancements with the integration of artificial intelligence (AI). AI techniques, particularly machine learning (ML) and deep learning, offer unprecedented opportunities to solve long-standing problems, identify patterns, and accelerate computations. This paper explores the applications, challenges, and future directions of AI in computational number theory, highlighting key achievements and areas of ongoing research. Computational number theory’s incorporation of artificial intelligence (AI) marks a revolutionary development in the investigation and solution of challenging mathematical issues. The computational complexity of issues like factorization, primality testing, and the creation of cryptographic algorithms has long presented a challenge to number theory, the area of mathematics dedicated to the study of integers and their properties. AI has the potential to revolutionize several fields with its strengths in automated reasoning, neural networks, and machine learning. AI’s ability to optimize algorithms, find new patterns, and resolve previously unsolvable issues has been shown in recent number theory applications. Large datasets of integers and their characteristics have been used to train machine learning models, which have led to notable advancements in prime number prediction and integer factorization methods. Additionally, modular forms and L- functions have been analyzed using deep learning, providing insight into intricate structures such as the Riemann Hypothesis. Reinforcement learning methods have also been used to automate the proof of theorems in fields like diophantine equations and elliptic curves and to find counterexamples to long- standing conjectures.

Number theory (or the theory of numbers [1]) starts from elementary (not necessarily simple and easy, and in fact, it is often very hard) number theory and grows up with analytic number theory (including additive and multiplicative number theory), algebraic number theory, geometric number theory (including arithmetic algebraic geometry), combinatorial number theory, computational number theory (including algorithmic number theory), to name just a few. Applied number theory, on the other hand, is involved in the application of various branches of number theory to a wide range of areas including, e.g., physics, chemistry, biology, graphics, arts, music, and particularly computing and digital communications [2]. Number theory was once viewed as the purest of the pure mathematics, with little application to other areas. However, with the advent of modern computers and digital communications, number theory becomes increasingly important and applicable to many areas ranging from natural sciences, engineering to social sciences.

The past few years have witnessed exciting discoveries in different areas of coding theory and computational number theory. The present monograph, which exhibits the author's Habilitationsschrift, is a collection of five different topics dealing with these two important fields. We will start with a purely coding theoretic question and finish with a discussion of some problems from computational number theory. Along the way, we will gradually change our focus from coding theory to number theory. Our emphasis is almost entirely on the development of fast and practical algorithms for the problems involved. In many cases it turns out that having a view for both number theory and coding theory is a clear advantage. This is best demonstrated by Chapters 2 and 3, where we encounter most of the interrelations between coding and number theory

Uses of Randomness in Algorithms and Protocols makes fundamental contributions to two different fields of complexity theory: computational number theory and cryptography. The most famous result is Goldwasser and Kilian's invention of a new approach to distinguish prime numbers from composites, using methods from the theory of elliptic curves over finite fields.The Goldwasser-Kilian algorithm is the first to yield a polynomial size proof of its assertions, ensuring correctness while still provably running fast on most inputs. This new primality test implies for the first time and without any assumptions that large certified primes can be generated in expected polynomial time under a distribution that is close to uniform. It provides a provocative new link between algebraic geometry and primality testing, one of the most ancient algorithmic problems in number theory. Heuristic implementations of the algorithm are currently considered to be the fastest existing methods to certify primes.Kilian also provides two elegant and original contributions to theoretical cryptography. He shows how to base general two-party protocols on a simple protocol, known as oblivious transfer, proving the first completeness result of this kind. He also introduces a generalization of interactive proof known as multi-prover interactive proof systems, and shows that anything provable in this model is provable in zero knowledge.Joe Kilian is a National Science Foundation Postdoctoral Fellow at MIT and Harvard.Contents: Introduction. New Techniques in Primality Testing. Committing Bits Using Oblivious Transfer. Circuit Evaluation Using Oblivious Transfer: The NC1 Circuit Base. ObliviousEvaluation of Arbitrary Circuits. Interactive Proof Systems with Multiple Provers.

This graduate textbook offers a self-contained introduction to the concepts and techniques of logarithmic geometry, a key tool for analyzing compactification and degeneration in algebraic geometry and number theory. It features a systematic exposition of the foundations of the field, from the basic results on convex geometry and commutative monoids to the theory of logarithmic schemes and their de Rham and Betti cohomology. The book will be of use to graduate students and researchers working in algebraic, analytic, and arithmetic geometry as well as related fields.

Developed from the authors popular graduate-level course, Computational Number Theory presents a complete treatment of number-theoretic algorithms. Avoiding advanced algebra, this self-contained text is designed for advanced undergraduate and beginning graduate students in engineering. It is also suitable for researchers new to the field and practitioners of cryptography in industry. Requiring no prior experience with number theory or sophisticated algebraic tools, the book covers many computational aspects of number theory and highlights important and interesting engineering applications. It first builds the foundation of computational number theory by covering the arithmetic of integers and polynomials at a very basic level. It then discusses elliptic curves, primality testing, algorithms for integer factorization, computing discrete logarithms, and methods for sparse linear systems. The text also shows how number-theoretic tools are used in cryptography and cryptanalysis. A dedicated chapter on the application of number theory in public-key cryptography incorporates recent developments in pairing-based cryptography. With an emphasis on implementation issues, the book uses the freely available number-theory calculator GP/PARI to demonstrate complex arithmetic computations. The text includes numerous examples and exercises throughout and omits lengthy proofs, making the material accessible to students and practitioners.

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.

Every second year, the GEOCRYPT conference brings together researchers from arithmetic geometry and cryptography. After Guadeloupe in 2009 and Corsica in 2011, French Polynesia was the host of its 2013 edition which took place in Punaauia, on the island of Tahiti, October 7-11. The main topic of GEOCRYPT has always been the application of pure mathematical techniques to the safety and efficiency of modern communication systems, with particular interest in the fields of arithmetic and algebraic geometry over finite fields, algorithms for finite fields, error correcting codes, cryptology, boolean functions, discrete dynamical systems, and their interactions. The 2013 edition had the honour to feature six invited and eleven contributed talks from renowned international experts presenting strong results recently obtained on topics ranging from pure mathematics to cryptographic algorithms. The GEOCRYPT 2013 Program Committee carefully evaluated the submitted abstracts and selected the best contributions for presentation at the conference.

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.

Abstract. Diophantine equations are mathematical equations that include only integer solutions and two or more unknown variables. To illustrate, the most elementary form of Diophantine equations are linear Diophantine equations when two different one-degree monomials added up to a constant value. Such equations were given the name of the extraordinary Greek mathematician Diophantus who made great contribution to algebra and number theory. In addition, they help in defining concepts in algebraic geometry and contribute to the development of algorithms for cryptography. They work in the same way as public keys do in cryptocurrencies within blockchain technology to curb cyber threats and scams targeting public systems. They serve as a means of securing transactions on a computer network when using cryptocurrencies such as bitcoins. These encryption techniques also permit cryptographers and mathematicians to create new ideas in relation to these number systems, thus making it possible for further cryptographic techniques to be put into place.

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions, and connects with representation theory, automorphic forms, Hodge theory, algebraic topology, and many other fields.

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions. An emphasis of the workshop was on p-adic techniques, but various other aspects including Hodge theory, Arakelov theory and global questions were discussed.

Arithmetic geometry is at the interface between algebraic geometry and number theory, and studies schemes over the ring of integers of number fields, or their p -adic completions. The talks covered a wide range of topics including the categorical Langlands program, Shimura varieties, complex and p -adic Hodge theory, homotopy theory, and Diophantine geometry.

Research in computational group theory, an active subfield of computational algebra, has emphasised three areas: finite permutation groups, finite solvable groups, and finitely presented groups. This book deals with the third of these areas. The author emphasises the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group. The work of Baumslag, Cannonito and Miller on computing nonabelian polycyclic quotients is described as a generalisation of Buchberger's Gröbner basis methods to right ideals in the integral group ring of a polycyclic group. Researchers in computational group theory, mathematicians interested in finitely presented groups and theoretical computer scientists will find this book useful.

Computational and algorithmic number theory are two very closely related subjects; they are both concerned with, among many others, computer algorithms, particularly efficient algorithms (including parallel and distributed algorithms, sometimes also including computer architectures), for solving different sorts of problems in number theory and in other areas, including computing and cryptography. Primality testing, integer factorization and discrete logarithms are, amongst many others, the most interesting, difficult and useful problems in number theory, computing and cryptography. In this chapter, we shall study both computational and algorithmic aspects of number theory. More specifically, we shall study various algorithms for primality testing, integer factorization and discrete logarithms that are particularly applicable and useful in computing and cryptography, as well as methods for many other problems in number theory, such as the Goldbach conjecture and the odd perfect number problem.