A Link to the Math. Connections Between Number Theory and Other Mathematical Topics
|title||A Link to the Math. Connections Between Number Theory and Other Mathematical Topics|
|publication type||Buch / Monographie / Herausgeberschaft|
|digital object identifier (doi)||DOI: 10.18442/805|
|note||Hildesheim, Diss. 2017|
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.