An experimental introduction to number theory [electronic resource] / Benjamin Hutz.

Machine generated contents note: ch. 1 Integers 1.The Integers and the Well Ordering Property 2.Divisors and the Division Algorithm 3.Greatest Common Divisor and the Euclidean Algorithm 4.Prime Numbers and Unique Factorization Exercises ch. 2 Modular Arithmetic 1.Basic Arithmetic 2.Inverses and Fermat's Little Theorem 3.Linear Congruences and the Chinese Remainder Theorem Exercises ch. 3 Quadratic Reciprocity and Primitive Roots 1.Quadratic Reciprocity 2.Computing mth Roots Modulo n 3.Existence of Primitive Roots Exercises ch. 4 Secrets 1.Basic Ciphers 2.Symmetric Ciphers 3.Diffie Hellman Key Exchange 4.Public Key Cryptography (RSA) 5.Hash Functions and Check Digits 6.Secret Sharing Exercises ch. 5 Arithmetic Functions 1.Euler Totient Function 2.Mobius Function 3.Functions on Divisors 4.Partitions Exercises ch. 6 Algebraic Numbers 1.Algebraic or Transcendental Note continued: 2.Quadratic Number Fields and Norms 3.Integers, Divisibility, Primes, and Irreducibles 4.Application: Sums of Two Squares Exercises ch. 7 Rational and Irrational Numbers 1.Diophantine Approximation 2.Height of a Rational Number 3.Heights and Approximations 4.Continued Fractions 5.Approximating Irrational Numbers with Convergents Exercises ch. 8 Diophantine Equations 1.Introduction and Examples 2.Working Modulo Primes 3.Pythagorean Triples 4.Fermat's Last Theorem 5.Pell's Equation and Fundamental Units 6.Waring Problem Exercises ch. 9 Elliptic Curves 1.Introduction 2.Addition of Points 3.Points of Finite Order 4.Integer Points and the Nagel-Lutz Theorem 5.Mordell Weil Group and Points of Infinite Order 6.Application: Congruent Numbers Exercises ch. 10 Dynamical Systems 1.Discrete Dynamical Systems 2.Dynatomic Polynomials 3.Resultant and Reduction Modulo Primes Note continued: 4.Periods Modulo Primes 5.Algorithms for Rational Periodic and Preperiodic Points Exercises ch. 11 Polynomials 1.Introduction to Polynomials 2.Factorization and the Euclidean Algorithm 3.Modular Arithmetic for Polynomials 4.Diophantine Equations for Polynomials Exercises.

Presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data.