Elementary Number Theory

Citation

Burton, D. M. (2011). Elementary Number Theory (7th ed.). McGraw-Hill.

Chapter Summary

The book has sixteen chapters, followed by miscellaneous problems, appendixes (general references, suggested further reading and tables), answers to selected problems and an index. Several chapters open with a section on a mathematician connected with the topic (Gauss, Fermat, Euler, Mersenne, Lagrange, Fibonacci, Ramanujan), which gives the book its historical thread.

Chapter 1: Preliminaries

Two sections, on mathematical induction and the binomial theorem, which prepare the proof techniques and identities the later chapters draw on.

Chapter 2: Divisibility Theory in the Integers

Opens with a section on early number theory, then develops the division algorithm, the greatest common divisor and the Euclidean algorithm, and ends with the linear Diophantine equation. This is the first chapter in which the central objects of the subject appear.

Chapter 3: Primes and Their Distribution

Covers the Fundamental Theorem of Arithmetic, the Sieve of Eratosthenes and the Goldbach conjecture. Primes and their distribution are treated before congruences, so the later chapters can refer to factorization throughout.

Chapter 4: The Theory of Congruences

After a section on Carl Friedrich Gauss, the chapter presents the basic properties of congruence, binary and decimal representations of integers, and linear congruences with the Chinese Remainder Theorem. The chapters on Fermat, Euler, primitive roots and reciprocity that follow all belong to the theory of congruences.

Chapter 5: Fermat's Theorem

Begins with Pierre de Fermat, then treats Fermat's Little Theorem together with pseudoprimes, Wilson's Theorem and the Fermat–Kraitchik factorization method.

Chapter 6: Number-Theoretic Functions

Discusses the sum and number of divisors, the Möbius inversion formula and the greatest integer function, and closes with an application to the calendar.

Chapter 7: Euler's Generalization of Fermat's Theorem

A section on Leonhard Euler leads to Euler's phi-function, Euler's Theorem and some further properties of the phi-function.

Chapter 8: Primitive Roots and Indices

Introduces the order of an integer modulo n, primitive roots for primes, the composite moduli that have primitive roots, and the theory of indices.

Chapter 9: The Quadratic Reciprocity Law

Develops Euler's criterion, the Legendre symbol and its properties, quadratic reciprocity itself and quadratic congruences with composite moduli. For many courses this is the destination of the classical sequence.

Chapter 10: Introduction to Cryptography

Runs from the Caesar cipher to public-key cryptography, then treats the knapsack cryptosystem and an application of primitive roots to cryptography. It is a three-section chapter, so it introduces the topic rather than treating it at length.

Chapter 11: Numbers of Special Form

Starting from Marin Mersenne, the chapter covers perfect numbers, Mersenne primes, amicable numbers and Fermat numbers.

Chapter 12: Certain Nonlinear Diophantine Equations

Two sections: one on a specific nonlinear Diophantine equation, and one on Fermat's Last Theorem.

Chapter 13: Representation of Integers as Sums of Squares

After a section on Joseph Louis Lagrange, the chapter treats sums of two squares and sums of more than two squares.

Chapter 14: Fibonacci Numbers

Introduces Fibonacci, the Fibonacci sequence and certain identities involving Fibonacci numbers.

Chapter 15: Continued Fractions

Beginning with Srinivasa Ramanujan, it covers finite and infinite continued fractions, Farey fractions and Pell's equation.

Chapter 16: Some Recent Developments

A survey chapter: Hardy, Dickson and Erdős; primality testing and factorization; an application of factoring to remote coin flipping; and the Prime Number Theorem and the zeta function.

Key Concepts

Chapters 1–3: Preliminaries, Divisibility, Primes

• Mathematical induction: the proof technique reviewed at the start and used for later arguments.
• Euclidean algorithm: the method for computing greatest common divisors that also solves linear Diophantine equations.
• Fundamental Theorem of Arithmetic: the unique factorization of integers into primes.

Chapters 4–5: Congruences and Fermat's Theorem

• Congruence: arithmetic modulo n, with linear congruences solved systematically.
• Chinese Remainder Theorem: solving several congruences with different moduli at once.
• Fermat's Little Theorem and Wilson's Theorem: the two classical results about powers and factorials modulo a prime.
• Pseudoprime: a composite number that passes a Fermat-type test, which links the theorem to primality testing.

Chapters 6–8: Arithmetic Functions, Euler's Theorem, Primitive Roots

• Möbius inversion formula: a way to recover a function from its summatory function over divisors.
• Euler's phi-function and Euler's Theorem: counting units modulo n and generalizing Fermat's theorem.
• Primitive roots and indices: numbers whose powers run through all the units modulo n, and the theory of indices built on them.

Chapters 9–10: Quadratic Reciprocity and Cryptography

• Legendre symbol and quadratic reciprocity: the tool for deciding whether a number is a square modulo a prime.
• Public-key cryptography: the chapter's route from classical ciphers to modern systems, with the knapsack cryptosystem as an example.

Chapters 11–16: Special Topics

• Perfect numbers, Mersenne primes and Fermat numbers: numbers of special form and their connections.
• Sums of squares: which integers are sums of two squares or of more than two.
• Continued fractions and Pell's equation: approximating real numbers and solving a classical Diophantine equation.
• Prime Number Theorem and zeta function: the analytic side of the subject, met only in the last section.

Critical Analysis

Strengths

• Historical context built into the structure. The publisher's description presents the book as a simple account of classical number theory set against a historical background running from antiquity to recent research. The table of contents bears this out: Gauss, Fermat, Euler, Mersenne, Lagrange, Fibonacci and Ramanujan each head a section, and Chapter 2 starts with early number theory.
• A clear classical spine. Chapters 2 to 9 move in a single line from divisibility through primes, congruences, Fermat and Euler, primitive roots and quadratic reciprocity. A reader can follow the standard course without detours, and the later chapters each take up a separate topic that can be chosen according to interest.
• Breadth of later material. Special numbers, sums of squares, Fibonacci numbers, continued fractions and a survey of recent developments give a course plenty of options once the core is done.
• Support for working problems. The book ends with miscellaneous problems, tables, answers to selected problems and lists of general references and further reading, so students can check work and find where to read next.

Limitations

• Classical scope. The contents list shows no algebraic number theory, and the Prime Number Theorem and zeta function appear only in the last section of the last chapter. Students heading toward algebraic or analytic number theory will need another text.
• Cryptography is brief. Cryptography has a single three-section chapter placed after quadratic reciprocity; courses centered on cryptography will want a book that integrates it earlier and more fully.
• Answers only to selected problems. The back matter provides answers to selected problems rather than to all of them, so a student working alone has limited means of checking proofs and will benefit from a study partner or instructor.
• Thin chapters. Several chapters have only a few sections: Chapter 12 has two, and Chapters 3, 10 and 14 have three each, so a course that wants depth on these topics will probably need to supplement them.

Where It Fits in the Course

Among the number theory books on this site, Burton is the classical one-semester text. Silverman's A Friendly Introduction to Number Theory starts from high-school algebra and emphasises experimenting and conjecturing, which makes it a gentler first book, and Robin Wilson's Number Theory: A Very Short Introduction is a compact tour with short proofs and history for readers who want an overview before committing to a full course. Burton is the full-length treatment of the classical syllabus, running through primitive roots and quadratic reciprocity to later chapters such as sums of squares and continued fractions.

Kraft and Washington's An Introduction to Number Theory with Cryptography covers much of the same classical ground but weaves cryptographic applications through it, so it is the natural alternative when security and computation are the goal. For a more demanding next step, Hardy and Wright's An Introduction to the Theory of Numbers is a classic treatise on elementary and analytic number theory, and Ireland and Rosen's A Classical Introduction to Modern Number Theory is a graduate text that assumes abstract algebra and leads to reciprocity laws, zeta functions over finite fields and elliptic curves.

How to Use This Book

The publisher writes the book for the one-semester undergraduate number theory course taken by mathematics majors, secondary education majors and computer science students. The site's prerequisite listing is some experience with proofs, including induction, and Chapter 1 reviews induction and the binomial theorem.

A practical plan for a first pass:

• Core: Chapters 2 to 5 and 7 to 9, from divisibility to quadratic reciprocity. Chapter 6 on number-theoretic functions (and its calendar application) can be taken or left according to time.
• Choose by interest: Chapter 10 for cryptography, Chapters 11 and 13 for special numbers and sums of squares, Chapter 15 for continued fractions and Pell's equation, and Chapter 16 as a preview of primality testing and analytic methods.
• Historical sections: read them for context; they introduce the people behind the results and are a good guide to why a topic is in the course.
• Problems: work a mix of computational exercises and proofs in each section, and use the answers to selected problems and the tables at the back to check results.

Readers who finish the book and want more can use its suggested further reading as a starting point, or move to one of the more advanced texts above.