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Preface To Everyone

Welcome to Number Theory! This book is an introduction to the theory and practice of the integers, especially positive integers – the numbers. We focus on connecting it to many areas of mathematics and dynamic, computer-assisted interaction. Let's explore!

Carl Friedrich Gauss, a great mathematician of the nineteenth century, is said to have quipped  1  that if mathematics is the queen of the sciences, then number theory is the queen of mathematics (hence the title of [E.5.4]). If you don't yet know why that might be the case, you are in for a treat.

In Wolfgang Sartorius von Waltershausen's rather lengthy and nearly hagiographic (‘his undying name ... whom no contemporary nation can place as an equal beside’) biography Gauss zum Gedächtnis; see the bottom of page 31 at the Internet Archive (link kindly provided by Neil McKay at the University of New Brunswick).

Number theory was (and is still occasionally) called ‘the higher arithmetic’, and that is truly where it starts. Even a small child understands that there is something interesting about adding numbers, and whether there is a biggest number, or how to put together fact families. Well before middle school many children will notice that some numbers don't show up in their multiplication tables much, or learn about factors and divisors. One need look no further than the excellent picture book You Can Count on Monsters [E.6.1] by Richard Evans Schwartz to see how compelling this can be.

Later on, perfect squares, basic geometric constructs, and even logarithms all can be considered part of arithmetic. Modern number theory is, at its heart, just the process of asking these same questions in more and more general situations, and more and more interesting situations.

They are situations with amazing depth. A sampling:

  • The question of what integers are possible areas of a right triangle seems very simple. Who could have guessed it would lead to fundamental advances in computer representation of elliptic curves?

  • There seems to be no nice formula for prime numbers, else we would have learned it in middle school. Yet who would have foreseen they are so very regular on average?

  • Taking powers of whole numbers and remainders while dividing are elementary and tedious operations. So why should taking remainders of tons of powers of whole numbers make online purchases more secure?

This book is designed to explore that fascinating world of whole numbers. It covers all the ‘standard’ questions, and perhaps some not-quite-as-standard topics as well. Roughly, it covers the following broad categories of topics.

  • Basic questions about integers

  • Basic congruence arithmetic

  • Units, primitive roots, and Euler's function (via groups)

  • Basics of cryptography, primality testing, and factorization

  • Integer and rational points on conic sections

  • The theory and practice of quadratic residues

  • Basics of arithmetic functions

  • The prime counting function and related matters

  • Connecting calculus to arithmetic functions

Finally, it won't take long to notice that the way in which this book is constructed emphasizes connections to other areas of math and encourages dynamic interaction. (See the note To the Instructor.) It is my hope that all readers will find this ‘in context and interactive’ approach enjoyable.