Fundamental Theorem of Arithmetic and the Division AlgorithmĪs the name rightly says, this theorem lies at the heart of all the concepts in number theory. But being the building blocks of arithmetic, these axioms are worth knowing.ġ. The fifth axiom is also popularly known as "principal of mathematical induction"īeing extremely basic, we would rarely need them directly, unless we want to prove every theorem in arithmetic from the first principles. (v) If a set contains the number 0 and it also contains the successor of every number in S, then S contains every natural number.
(iv) Different natural numbers have different successors
(iii) 0 is not the successor of any natural number (ii) Every natural number has a successor, which is also a natural number The entire formalization of arithmetic is based on five fundamental axioms, called Peano axioms, which define properties of natural numbers. External references (mostly from Wikipedia and Wolfram) have been provided at many places for further details. Rather, this writeup is intended to act as a reference. It is neither an introductory tutorial, nor any specific algorithms are discussed here. Upper-level undergraduates, graduates and researchers in the field of number theory will find this book to be a valuable resource.This writeup discusses few most important concepts in number theory that every programmer should ideally know.
In the last chapter they review several further applications of number theory, ranging from check-digit systems to quantum computation and the organization of raster-graphics memory. Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters 2-5 and offer a glimpse of advanced results that are presented without proofs and require more advanced mathematical skills. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc. Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas. This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory.
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