A Friendly Introduction to Number Theory

Citation

Silverman, J. H. (2017). A Friendly Introduction to Number Theory (4th ed., Classic Version). Pearson. (Fourth edition first published 2012.)

Chapter Summary

The fourth edition has 46 printed chapters, most of them short (often five to ten pages), plus four further chapters and two appendices published free online. The author's site gives the printed length as ix + 409 pages with 56 online pages. An introduction includes a guide for instructors and a discussion of computers, and a flowchart on page ix shows how the chapters depend on each other. The book grew out of Brown University's Math 42, a course designed to bring non-science majors to college mathematics.

Chapters 1–4: What Is Number Theory?; Pythagorean Triples; Pythagorean Triples and the Unit Circle; Sums of Higher Powers and Fermat's Last Theorem

The book opens with a gallery of number types (squares, primes, triangular, perfect and Fibonacci numbers) and a tour of typical questions about them. It then classifies Pythagorean triples, first algebraically and then geometrically through rational points on the unit circle, and asks what happens for higher powers, leading to Fermat's Last Theorem.

Chapters 5–7: Divisibility and the Greatest Common Divisor; Linear Equations and the Greatest Common Divisor; Factorization and the Fundamental Theorem of Arithmetic

The Euclidean algorithm, solving linear equations ax + by = c, and unique factorization into primes.

Chapters 8–11: Congruences; Congruences, Powers, and Fermat's Little Theorem; Congruences, Powers, and Euler's Formula; Euler's Phi Function and the Chinese Remainder Theorem

Arithmetic modulo m, including proof by contradiction and the fact that a polynomial of degree d has at most d roots modulo p. These chapters cover Fermat's little theorem, Euler's formula, the phi function and the Chinese Remainder Theorem.

Chapters 12–15: Prime Numbers; Counting Primes; Mersenne Primes; Mersenne Primes and Perfect Numbers

The infinitude of primes (Chapter 1 promises proofs for primes congruent to 1 and 3 modulo 4 in Chapters 12 and 21, with a discussion of Dirichlet's general theorem), counting primes, Mersenne primes and their link with perfect numbers.

Chapters 16–19: Powers Modulo m and Successive Squaring; Computing kth Roots Modulo m; Powers, Roots, and "Unbreakable" Codes; Primality Testing and Carmichael Numbers

Fast modular exponentiation and the computation of roots modulo m lead to the RSA public-key cryptosystem in Chapter 18, followed by primality tests and Carmichael numbers.

Chapters 20–23: Squares Modulo p; Is −1 a Square Modulo p? Is 2?; Quadratic Reciprocity; Proof of Quadratic Reciprocity

Quadratic residues and Legendre symbols, the special cases −1 and 2, the law of quadratic reciprocity (with Jacobi symbols), and Eisenstein's proof. The preface marks Chapter 23 as significantly more difficult and optional.

Chapters 24–25: Which Primes Are Sums of Two Squares?; Which Numbers Are Sums of Two Squares?

Fermat's theorem on primes that are sums of two squares, and its extension to all integers.

Chapters 26–29: As Easy as One, Two, Three; Euler's Phi Function and Sums of Divisors; Powers Modulo p and Primitive Roots; Primitive Roots and Indices

A chapter on mathematical induction, new in the fourth edition, then sums of divisors. Primitive roots and indices follow, with an application to Costas arrays. They were moved later in this edition because students find the Primitive Root Theorem among the hardest in the book.

Chapters 30–34: The Equation X⁴ + Y⁴ = Z⁴; Square–Triangular Numbers Revisited; Pell's Equation; Diophantine Approximation; Diophantine Approximation and Pell's Equation

A proof that no fourth power is a sum of two fourth powers, then square-triangular numbers leading to Pell's equation, which is solved with the help of Diophantine approximation.

Chapters 35–37: Number Theory and Imaginary Numbers; The Gaussian Integers and Unique Factorization; Irrational Numbers and Transcendental Numbers

Gaussian integers and their unique factorization, and a chapter on irrational and transcendental numbers.

Chapters 38–40: Binomial Coefficients and Pascal's Triangle; Fibonacci's Rabbits and Linear Recurrence Sequences; Oh, What a Beautiful Function

Binomial coefficients and Pascal's triangle, Fibonacci and other linear recurrence sequences (including the period of the Fibonacci sequence modulo p), and a chapter that introduces big-Oh notation and uses calculus.

Chapters 41–46: Cubic Curves and Elliptic Curves; Elliptic Curves with Few Rational Points; Points on Elliptic Curves Modulo p; Torsion Collections Modulo p and Bad Primes; Defect Bounds and Modularity Patterns; Elliptic Curves and Fermat's Last Theorem

An introduction to elliptic curves: rational points, points modulo p, bad primes, defect bounds and patterns related to modularity. The sequence ends with a sketch of some of the ideas in Wiles's proof of Fermat's Last Theorem.

Chapters 47–50 (online): The Topsy-Turvy World of Continued Fractions; Continued Fractions and Pell's Equation; Generating Functions; Sums of Powers

Free online chapters on continued fractions and their use for Pell's equation, generating functions and sums of powers, with appendices on factorizations of small integers and a list of primes.

Key Concepts

Chapters 1–11: Foundations

• Pythagorean triples and rational points: integer solutions of a² + b² = c² correspond to rational points on the unit circle.
• Euclidean algorithm: computing greatest common divisors and solving linear equations in integers.
• Fundamental Theorem of Arithmetic: unique factorization into primes.
• Congruences, Fermat's little theorem and Euler's formula: the basic tools of modular arithmetic.
• Chinese Remainder Theorem: solving simultaneous congruences with coprime moduli.

Chapters 12–19: Primes and cryptography

• Mersenne primes and perfect numbers: primes of the form 2ⁿ − 1 and their link to even perfect numbers.
• Successive squaring and kth roots modulo m: the computations behind RSA.
• Carmichael numbers and primality tests: composites that fool Fermat's test, and stronger tests.

Chapters 20–29: Quadratic residues and primitive roots

• Quadratic reciprocity: the law relating whether p is a square modulo q to whether q is a square modulo p.
• Sums of two squares: which primes and integers can be written as a² + b².
• Primitive root: a number whose powers give every nonzero residue modulo p.

Chapters 30–46: Diophantine equations and beyond

• Pell's equation: x² − Dy² = 1, solved through Diophantine approximation.
• Gaussian integers: complex numbers a + bi with integer a, b, which also have unique factorization.
• Elliptic curve: a cubic curve whose points can be studied over the rationals and modulo p, central to the proof of Fermat's Last Theorem.

Critical Analysis

Strengths

• Discovery before proof. The preface states the goal of leading readers to think mathematically: numerical examples are examined for patterns and used to make conjectures, and only then are theorems proved. Chapters 1–6, which the author makes freely available, follow this pattern, with experiments followed by proofs.
• Minimal prerequisites, real mathematics. Only high-school algebra is required. The author notes that calculus appears in an essential way only in Chapters 40 and 49, which can be omitted. Yet the book reaches quadratic reciprocity, Pell's equation, Gaussian integers and elliptic curves.
• Built for flexible courses. The introduction says the book holds about two semesters of material. It recommends Chapters 1–11 as the basis and continuing to RSA in Chapter 18. It then offers several one-semester routes, including one designed for students heading to computer science or cryptography. The dependency flowchart and the optional status of Chapter 23 make selection easy.
• Open-ended exercises and sensible computing. Many exercises are deliberately open-ended ("Tell me as much as you can about…"); the author warns that students may find this unsettling at first. The author encourages students to program the algorithms themselves rather than rely on packaged software, and marks computer exercises and projects.

Limitations

• Short chapters mean many chapters. With 46 printed chapters and some topics spread across several, the book can feel fragmented. Instructors need the flowchart to plan a route.
• Pace may feel slow to strong students. The informal, example-led style suits newcomers. Students who already know congruences may find the early chapters leisurely, and the book is not a compact reference.
• Some material is online only. Continued fractions, generating functions and sums of powers were moved out of the printed book. The author notes that some low-cost printings omitted Chapter 21 and renumbered later chapters, and provides that chapter free online.

Where It Fits in the Course

Silverman's book is the friendly first textbook in this number theory list. Robin Wilson's Number Theory: A Very Short Introduction is a quick overview that can be read beforehand to see the landscape. Silverman then teaches students to do the mathematics. David Burton's Elementary Number Theory is a more conventional undergraduate textbook with greater historical and computational breadth. Kraft and Washington's An Introduction to Number Theory with Cryptography suits students who want cryptography more central than Silverman's Chapters 16–19.

Ireland and Rosen's A Classical Introduction to Modern Number Theory and Hardy and Wright's An Introduction to the Theory of Numbers are the natural next steps. Ireland and Rosen continues into algebraic number theory and Hardy and Wright is the classic comprehensive treatise. Silverman's chapters on Gaussian integers and elliptic curves are a gentle preview of the topics Ireland and Rosen develop formally.

How to Use This Book

The book is written for undergraduates in a first number theory course, including non-majors, and for independent study. The prerequisites are high-school algebra and a willingness to experiment; a simple calculator is enough. Those who can program are encouraged to implement the algorithms.

• Core: Chapters 1–11, then continue to RSA in Chapter 18, as the author recommends.
• Choose a second half: the introduction suggests routes such as quadratic reciprocity, sums of squares, Pell's equation and continued fractions (Chapters 20–26, 31–34, 47–48), or Fermat's equation and elliptic curves (Chapters 30–34, 41–46). A computer-science route is Chapters 19–22, 26–29 and 38–40. Chapter 23 may be skipped.
• Work the open-ended exercises. Collect data, try special cases and look for generalizations before seeking a proof.
• Use the author's website for the online chapters, the errata list and data for the longer computer exercises.