A Friendly Introduction to Number Theory by Joseph Silverman
A Friendly Introduction to Number Theory - Table of Contents
1. What Is Number Theory?
2. Pythagorean Triples
3. Pythagorean Triples and the Unit Circle
4. Sums of Higher Powers and Fermat’s Last Theorem
5. Divisibility and the Greatest Common Divisor
6. Linear Equations and the Greatest Common Divisor
7. Factorization and the Fundamental Theorem of Arithmetic
8. Congruences
9. Congruences, Powers, and Fermat’s Little Theorem
10. Congruences, Powers, and Euler’s Formula
11. Euler’s Phi Function and the Chinese Remainder Theorem
12. Prime Numbers
13. Counting Primes
14. Mersenne Primes
15. Mersenne Primes and Perfect Numbers
16. Powers Modulo m and Successive Squaring
17. Computing k-th Roots Modulo m
18. Powers, Roots, and “Unbreakable” Codes
19. Primality Testing and Carmichael Numbers
20. Squares Modulo p
21. Is -1 a Square Modulo p? Is 2?
22. Quadratic Reciprocity
23. Proof of Quadratic Reciprocity
24. Which Primes Are Sums of Two Squares?
25. Which Numbers Are Sums of Two Squares?
26. As Easy as One, Two, Three
27. Euler’s Phi Function and Sums of Divisors
28. Powers Modulo p and Primitive Roots
29. Primitive Roots and Indices
30. The Equation X4 + Y4 = Z4
31. Square–Triangular Numbers Revisited
32. Pell’s Equation
33. Diophantine Approximation
34. Diophantine Approximation and Pell’s Equation
35. Number Theory and Imaginary Numbers
36. The Gaussian Integers and Unique Factorization
37. Irrational Numbers and Transcendental Numbers
38. Binomial Coefficients and Pascal’s Triangle
39. Fibonacci’s Rabbits and Linear Recurrence Sequences
40. Oh, What a Beautiful Function
41. Cubic Curves and Elliptic Curves
42. Elliptic Curves with Few Rational Points
43. Points on Elliptic Curves Modulo p
44. Torsion Collections Modulo p and Bad Primes
45. Defect Bounds and Modularity Patterns
46. Elliptic Curves and Fermat’s Last Theorem
47. The Topsy-Turvy World of Continued Fractions [online]
48. Continued Fractions and Pell’s Equation [online]
49. Generating Functions [online]
50. Sums of Powers [online]
What You Will Learn in A Friendly Introduction to Number Theory
"A Friendly Introduction to Number Theory" by Joseph H. Silverman is a welcoming and engaging entry point into "number theory" for beginners. Written with clarity and warmth, the book assumes only basic algebra and gradually introduces readers to deeper mathematical thinking. Through simple examples and patterns, it helps students develop intuition while learning how mathematicians approach problems and ideas.
The book focuses strongly on "proof techniques", encouraging readers to experiment, make conjectures, and justify results logically. Core topics include "divisibility", "congruences", prime numbers, and modular arithmetic, with natural connections to real-world ideas such as "cryptography". Rather than presenting results as facts to memorize, Silverman guides readers to discover why statements are true, building confidence and problem-solving skills along the way.
Ideal for undergraduates, self-learners, and first courses in number theory, this book balances rigor with accessibility. Its friendly tone, thoughtful exercises, and emphasis on mathematical thinking make it especially effective for students transitioning from computational math to abstract reasoning. By the end, readers gain both a solid foundation in "number theory" and a deeper appreciation for how mathematical ideas are explored, tested, and proven.
Book Details & Specifications
Title:
A Friendly Introduction to Number Theory by Joseph Silverman
Publisher:
Pearson
Year:
2012
Pages:
456
Type:
PDF
Language:
English
ISBN-10 #:
0134689461
ISBN-13 #:
978-0134689463
License:
External Educational Resource
Amazon:
Amazon
About the Author: Joseph H. Silverman
The author
Joseph H. Silverman
is an American mathematician and professor at "Brown University", widely known for his work in "number theory" and arithmetic geometry. He earned his PhD from Harvard University and has made major contributions to elliptic curves and Diophantine equations. Silverman is also respected for writing clear, student-friendly mathematics textbooks. His work, "A Friendly Introduction to Number Theory", introduces core ideas such as "primes", "modular arithmetic", and Diophantine problems using intuitive explanations and engaging examples.
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