# A Brief Journey in Discrete Mathematics

A Brief Journey in Discrete Mathematics by Randolph Nelson

## Contents of Brief Journey in Discrete Mathematics

• Introduction
• Let Me Count the Ways
• Permutations: With and Without Replacement
• Dearrangements
• Combinations: Without Replacement
• Binomial Identities
• Combinations with Replacement
• Binomial-R Identities
• Polynomial Solutions to Combinatorial
• Problems
• Transforms and Identities
• Syntax Precedes Semantics
• Stirling Numbers of the First Kind
• Stirling Numbers of the Second Kind
• The Stirling Transform and Inverse
• Combinatorial Interpretation
• Fearful Symmetry
• Symmetric Functions
• Simple Polynomials
• The Quadratic Equation
• Equation of the Minimum Distance Line
• The Pythagorean Theorem
• Cubic Polynomials
• Elementary Symmetric Polynomials
• Newton–Girard Formula
• Identities and Combinatorial Coefficients
• Inclusion–Exclusion
• Fundamental Theorem of Symmetric Polynomials
• Galois’ Theorem and Numerical Solutions
• All That Glitters Is Not Gold
• The Golden Ratio
• Fibonacci Numbers
• A Closed Form Solution
• An Alternate Derivation
• Generalized Fibonacci Numbers
• k-Bonacci Numbers
• Generalization of the Fibonacci Recurrence
• Heads I Win, Tails You Lose
• The Mathematical Model
• Games That End Even
• Catalan Numbers
• Non-intuitive Results
• The Correct Insight
• Particular Sequences
• Conclusions
• Sums of the Powers of Successive Integers
• A General Equation
• Iterative Approach
• Triangular Numbers
• Cauchy’s Theorem
• As Simple as + =
• Modular Arithmetic
• Fermat’s Little Theorem
• Lagrange’s Theorem
• Wilson’s Theorem
• Cryptography
• Hidden in Plain Sight
• Properties of Prime Numbers
• Properties of Integer Divisors
• The Prime Counting Function
• There Is Always a Prime Between n and n
• The Prime Number Theorem with a
• Controversy
• Running Off the Page
• Simple Continued Fractions
• Periodic Simple Continued Fractions
• Summary of Results
• General Method to Create a Continued Fraction
• Integer Quadratics and Quadratic Surds
• Approximations Using Continued Fractions
• Best Approximations
• Lagrange’s Theorem and Historical Review
• A Tools of the Trade
• A Recurrence Relationships
• A Adding Zero to an Equation
• A Induction
• A Contradiction
• A Order of Summations
• B Notation and Identities Derived in the Book

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