Friendly Introduction to Number Theory, A (Classic Version)
$133.32
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Description
About our author
Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and 7 books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography. He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.
A Friendly Introduction to Number Theory, 4th Edition introduces students to the overall themes and methodology of mathematics through the detailed study of one particular facet–number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.
Preface
Flowchart of Chapter Dependencies
Introduction
- What Is Number Theory?
- Pythagorean Triples
- Pythagorean Triples and the Unit Circle
- Sums of Higher Powers and Fermat’s Last Theorem
- Divisibility and the Greatest Common Divisor
- Linear Equations and the Greatest Common Divisor
- Factorization and the Fundamental Theorem of Arithmetic
- Congruences
- Congruences, Powers, and Fermat’s Little Theorem
- Congruences, Powers, and Euler’s Formula
- Euler’s Phi Function and the Chinese Remainder Theorem
- Prime Numbers
- Counting Primes
- Mersenne Primes
- Mersenne Primes and Perfect Numbers
- Powers Modulo m and Successive Squaring
- Computing kth Roots Modulo m
- Powers, Roots, and “Unbreakable” Codes
- Primality Testing and Carmichael Numbers
- Squares Modulo p
- Is -1 a Square Modulo p? Is 2?
- Quadratic Reciprocity
- Proof of Quadratic Reciprocity
- Which Primes Are Sums of Two Squares?
- Which Numbers Are Sums of Two Squares?
- As Easy as One, Two, Three
- Euler’s Phi Function and Sums of Divisors
- Powers Modulo p and Primitive Roots
- Primitive Roots and Indices
- The Equation X4 + Y4 = Z4
- Square – Triangular Numbers Revisited
- Pell’s Equation
- Diophantine Approximation
- Diophantine Approximation and Pell’s Equation
- Number Theory and Imaginary Numbers
- The Gaussian Integers and Unique Factorization
- Irrational Numbers and Transcendental Numbers
- Binomial Coefficients and Pascal’s Triangle
- Fibonacci’s Rabbits and Linear Recurrence Sequences
- Oh, What a Beautiful Function
- 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
- The Topsy-Turvey World of Continued Fractions [online]
- Continued Fractions, Square Roots, and Pell’s Equation [online]
- Generating Functions [online]
- Sums of Powers [online]
Further Reading
Index
A. Factorization of Small Composite Integers [online]
B. A List of Primes [online]
For one-semester undergraduate courses in Elementary Number Theory
This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.
A Friendly Introduction to Number Theory, 4th Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet–number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
New and updated features of this title
- Many new exercises appear throughout the text.
- Content updates throughout include:
- A new chapter on mathematical induction (Chapter 26).
- Some material on proof by contradiction has been moved forward to Chapter 8.
- The chapters on primitive roots (Chapters 28–29) have been moved to follow the chapters on quadratic reciprocity and sums of squares (Chapters 20–25).
- Chapter 22 now includes a proof of part of quadratic reciprocity for Jacobi symbols, with the remaining parts included as exercises.
- Quadratic reciprocity is now proved in full. The proofs for (-1/p) and (2/p) remain as before in Chapter 21, and there is a new chapter (Chapter 23) that gives Eisenstein’s proof for (p/q)(q/p). Chapter 23 is significantly more difficult than the chapters that precede it, and it may be omitted without affecting the subsequent chapters.
- As an application of primitive roots, Chapter 28 discusses the construction of Costas arrays.
- Chapter 39 includes a proof that the period of the Fibonacci sequence modulo p divides p – 1 when p is congruent to 1 or 4 modulo 5.
Hallmark features of this title
- 50 short chapters provide flexibility and options for instructors and students. A flowchart of chapter dependencies is included.
- 5 basic steps are emphasized throughout the text to help readers develop a robust thought process: Experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.
- RSA cryptosystem, elliptic curves, and Fermat’s Last Theorem are featured, showing the real-life applications of mathematics.
Additional information
| Dimensions | 1.05 × 6.00 × 8.95 in |
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| Subjects | mathematics, higher education, number theory, Calculus, Applied & Advanced Math, Advanced Math |
