Number TheoryIn Context and Interactive

Karl-Dieter Crisman

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Front Matter

  • Copyright Information
  • To Everyone
  • To the Student
  • To the Instructor
  • About the Author
  • Dedication
  • Acknowledgements

1Prologue

  • A first problem
  • Review of previous ideas
  • Where are we going?
  • Exercises
  • How to use computation

2Basic Integer Division

  • The Division Algorithm
  • The GCD
  • The Euclidean Algorithm
  • The Bezout Identity
  • Exercises

3From Linear Equations to Geometry

  • Linear Diophantine Equations
  • Geometry of Equations
  • Positive Integer Lattice Points
  • Pythagorean Triples
  • Surprises in Integer Equations
  • Exercises
  • Two facts from the gcd

4Prime Time

  • To Infinity and Beyond
  • The Fundamental Theorem of Arithmetic
  • Exercises

5First Steps with Congruence

  • Introduction to Congruence
  • Going Modulo First
  • Properties of Congruence
  • Equivalence classes
  • Toward Congruences
  • Exercises

6Linear Congruences

  • Solving Linear Congruences
  • A bigger strategy
  • Systems of Linear Congruences
  • Using the Chinese Remainder Theorem
  • More complicated cases
  • Exercises

7Important Congruence Theorems

  • Prime fun and Square Roots
  • From Linear to General?
  • Wilson's Theorem and Fermat's Little Theorem
  • Polynomials and Lagrange's Theorem
  • Epilogue: Why congruences matter
  • Exercises

8The Group of Integers Modulo \(n\)

  • The integers modulo \(n\)
  • Powers
  • Essential Facts about Groups for Number Theory

9The Group of Units and the Euler \(\phi\) Function

  • Groups and Number Systems
  • The Euler Phi Function
  • Using Euler's Theorem
  • Exploring Euler's Function
  • Proofs and reasons
  • Exercises

10Primitive Roots

  • Primitive Roots
  • A better way to primitive roots
  • When is there a primitive root?
  • Prime numbers have primitive roots
  • A practical use of primitive roots
  • Exercises

11An Introduction to Cryptography

  • What is cryptography?
  • Encryption
  • A modular exponentiation cipher
  • An interesting application: key exchange
  • RSA Public Key
  • RSA and (lack of) security
  • Exercises

12Some theory behind cryptographic practice

  • Finding More Primes
  • Primes — Probably
  • Another primality test
  • Strong Pseudoprimes
  • Introduction to Factorization
  • A Taste of Modernity
  • Exercises

13Sums of Squares

  • Some First Ideas
  • Primes can be written in at most one way
  • A Lemma about Square Roots Modulo \(n\)
  • Primes as Sum of Squares
  • Proving sums of squares
  • All the squares fit to be summed
  • A One-Sentence Proof
  • Exercises

14Beyond Sums of Squares

  • A Complex Situation
  • More sums of squares
  • Related Questions about Sums
  • Exercises

15Points on Curves

  • Rational Points on Conics
  • More about points, rational and integer
  • Bachet and Mordell curves
  • More on Mordell
  • Points on Quadratic Curves
  • Making more and more and more points
  • The algebraic story
  • Exercises

16Solving Quadratic Congruences

  • Square Roots
  • General quadratic congruences
  • Quadratic residues
  • Send in the Groups
  • Euler's Criterion
  • The Legendre Symbol
  • Our First Full Computation
  • Exercises

17Quadratic Reciprocity

  • More Legendre Symbols
  • Another Criterion
  • Using Eisenstein's Criterion
  • Quadratic Reciprocity
  • Some Surprising Applications of QR
  • A Proof of Quadratic Reciprocity
  • Exercises

18An Introduction to Functions

  • Three questions
  • Three questions, again
  • Exercises

19Counting and Summing Divisors

  • Exploration: a new sequence of functions
  • Conjectures and proofs
  • The size of the sum of divisors function
  • Perfect Numbers
  • More regarding perfection
  • Odd Perfect Numbers
  • Exercises

20Long-Term Function Behavior

  • Sums of squares, once more
  • Average of tau
  • Digging deeper and finding limits
  • Heuristics for the sum of divisors
  • Looking ahead
  • Exercises

21The Prime Counting Function

  • First Steps
  • Some history
  • The Prime Number Theorem
  • A slice of the Prime Number Theorem
  • Exercises

22More on Prime Numbers

  • Prime Races
  • Sequences and Primes
  • Types of Primes
  • Exercises

23New Functions from Old

  • The Moebius function
  • New Arithmetic Functions from Old
  • Making new functions
  • Generalizing Moebius
  • Exercises

24Infinite Sums and Products

  • Products and Sums
  • The Riemann Zeta Function
  • From Riemann to Dirichlet and Euler
  • Multiplication
  • Multiplication and Inverses
  • Four Facts
  • Exercises

25Further Up and Further In

  • Taking the PNT further
  • Improving the PNT
  • Toward the Riemann Hypothesis
  • Connecting to the Primes
  • Connecting to zeta
  • Connecting to zeros
  • The Riemann Explicit Formula
  • Epilogue
  • Exercises
Authored in MathBook XML

BookNumber Theory

Welcome to Number Theory! This course and 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!

Front Matter1Prologue2Basic Integer Division3From Linear Equations to Geometry4Prime Time5First Steps with Congruence6Linear Congruences7Important Congruence Theorems8The Group of Integers Modulo \(n\)9The Group of Units and the Euler \(\phi\) Function10Primitive Roots11An Introduction to Cryptography12Some theory behind cryptographic practice13Sums of Squares14Beyond Sums of Squares15Points on Curves16Solving Quadratic Congruences17Quadratic Reciprocity18An Introduction to Functions19Counting and Summing Divisors20Long-Term Function Behavior21The Prime Counting Function22More on Prime Numbers23New Functions from Old24Infinite Sums and Products25Further Up and Further In