Discovering New LL-Function Relations Using Algebraic Sieving

We report the discovery of new results relating $L$-functions, which typically encode interesting information about mathematical objects, obtained in a \emph{semi-automated} fashion using an algebraic sieving technique. Algebraic sieving initially comes from cryptanalysis, where it is used to solve factorization, discrete logarithms, or to produce signature forgeries in cryptosystems such as RSA. We repurpose the technique here to provide candidate identities, which can be tested and ultimately formally proven. A limitation of our technique is the need for human intervention in the post-processing phase, to determine the most general form of conjectured identities, and to provide a proof for them. Nevertheless we report 29 identities that hitherto never appeared in the literature, 9 of which we could completely prove, the remainder being numerically valid over all tested values. This work complements other instances in the literature where this type of automated symbolic computation has served as a productive step toward theorem proving; it can be extremely helpful in figuring out what it is that one should attempt to prove

A Pythagorean Introduction to Number Theory : Right Triangles, Sums of Squares, and Arithmetic

In the ?rst section of this opening chapter we review two different proofs of the PythagoreanTheorem,oneduetoEuclidandtheotheroneduetoaformerpresident oftheUnitedStates,JamesGar?eld.Inthesamesectionwealsoreviewsomehigher dimensional analogues of the Pythagorean Theorem. Later in the chapter we de?ne Pythagorean triples; explain what it means for a Pythagorean triple to be primitive; and clarify the relationship between Pythagorean triples and points with rational coordinates on the unit circle. At the end we list the problems that we will be interested in studying in the book. In the notes at the end of the chapter we talk about Pythagoreans and their, sometimes strange, beliefs. We will also brie?y review the history of Pythagorean triples

A Link to the Math. Connections Between Number Theory and Other Mathematical Topics

Number theory is one of the oldest mathematical areas. This is perhaps one of the reasons why there are many connections between number theory and other areas inside mathematics. This thesis is devoted to some of those connections. In the first part of this thesis I describe known connections between number theory and twelve other areas, namely analysis, sequences, applied mathematics (i.e., probability theory and numerical mathematics), topology, graph theory, linear algebra, geometry, algebra, differential geometry, complex analysis, physics and computer science, and algebraic geometry. We will see that the concepts will not only connect number theory with these areas but also yield connections among themselves. In the second part I present some new results in four topics connecting number theory with computer science, graph theory, algebra, and linear algebra and analysis, respectively. [...] In the next topic I determine the neighbourhood of the neighourhood of vertices in some special graphs. This problem can be formulated with generators of subgroups in abelian groups and is a direct generalization of a corresponding result for cyclic groups. In the third chapter I determine the number of solutions of some linear equations over factor rings of principal ideal domains R. In the case R = Z this can be used to bound sums appearing in the circle method. Lastly I investigate the puzzle “Lights Out” as well as variants of it. Of special interest is the question of complete solvability, i.e., those cases in which all starting boards are solvable. I will use various number theoretical tools to give a criterion for complete solvability depending on the board size modulo 30 and show how this puzzle relates to algebraic number theory

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout