Number and Proof

An Introduction to Number Theory and its Foundations
Erscheinungsjahr 2027
ISBN: 978-3-032-41932-3
Verlag: Springer

Buch, Englisch, 217 Seiten, Format (B × H): 155 mm x 235 mm

Reihe: Springer Undergraduate Mathematics Series

invites readers on a rigorous and engaging journey from the most familiar objects in mathematics, the natural numbers, to some of the central ideas of modern number theory. Written for undergraduate students in mathematics, science, and philosophy, the book develops proof-writing skills through active exploration while asking deeper questions: What are numbers? What does it mean to prove something? And how certain can mathematical knowledge really be? Drawing on history, logic, and philosophy as well as mathematics itself, the text illuminates the foundations underlying arithmetic and mathematical reasoning.

Beginning with the Peano axioms and the construction of the integers, the book guides readers carefully through the development of rigorous arguments before introducing the classic results of elementary number theory. Along the way, students encounter prime factorization, modular arithmetic, Diophantine equations, quadratic reciprocity, and finally the RSA public-key cryptosystem. Extensive exercises, many designed to encourage discovery and conjecture before formal proof, make the text equally suitable for classroom use and independent study.

Distinctive in both scope and style, the text treats mathematics not merely as a collection of results but as a human intellectual enterprise shaped by millennia of debate about logic, certainty, and the nature of knowledge. By connecting foundational questions with beautiful mathematical ideas and practical applications, the author offers readers an accessible introduction to proof, a rewarding encounter with number theory, and an invitation to explore the wider landscape of mathematics.

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Susanka, Lawrence

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

Preface.- Introduction.- 1 The Peano Axioms.- 2 The Integers: Their Order and Arithmetic.- 3 The Basic Results of Elementary Number Theory.- 4 Polynomial Congruences.- 5 Quadratic Reciprocity.- 6 Remarks on Diophantine Equations with More Variables or Higher Degree.- 7 Public Key Encryption.- Appendices A-H.- References.- Index of Symbols.- Index of Names.- Index.

Über Autor(innen)

Lawrence Susanka received his PhD in Mathematics in 1991 from the University of Minnesota. He was tenured at Bellevue College for 28 years until 2019. He currently helps manage the Math and Physics Colloquium series at Bellevue. Dr. Susanka has an active interest in physics and mathematics, rock and ice climbing, chess problems, horses and running mountain trails. He is a supporter of The Planetary Society, various skeptical organizations such as the Committee for Skeptical Inquiry, and is a fan of science fiction in its myriad forms. Professor Susanka is the author of Linear Algebra in Context, (c) 2025, ISBN: 978-3-031-78423-1.