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Elementary Number Theory

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Successful completion of Linear Algebra and Introduction to Abstract Math or the equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit or one semester of college credit equal to or greater than an AP class

Course Code: ENT

Course Description

Description

Elementary Number Theory gives advanced students an introduction to the deep theory of the integers, with focus on the properties of prime numbers, and integer or rational solutions to equations. This course covers topics similar to the third year undergraduate, in-person Elementary Number Theory course at Johns Hopkins University. This course focuses on detailed exploration of topics as well as proof techniques.  Historical background for various problems will be provided throughout the course. Prime numbers and elliptic curves are studied with applications to cryptography.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include homework, chapter exams, and a cumulative midterm and final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

Topics covered include:

  • Divisibility
  • Unique Factorization 
  • Congruences 
  • Number-Theoretic Functions 
  • Primitive Roots 
  • Diophantine Equations 
  • Continued Fractions 
  • Quadratic Residues 
  • Distribution of Primes

For a detailed list of topics, click the List of Topics tab.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns.

Videos from YouTube or other web providers may be present in the course. Video recommendations or links provided at end of videos are generated by the video host provider and are not CTY recommendations.

Proctor Requirements

For enrollment starting on or after January 1, 2019, a proctor is required for this course if the student’s goal is to obtain a grade. Please review Proctor Requirements for more details.

Materials Needed

A textbook purchase is required for this course:

A Friendly Introduction to Number Theory, 4th ed., by Joseph H. Silverman

  • ISBN-13: 978-0321816191
  • ISBN-10: 0321816196

List of Topics

Unit 1 – Introduction

  • What is Number Theory?
  • As Easy as One, Two, Three
  • Pythagorean Triples
  • Pythagorean Triples and the Unit Circle
  • Sums of Higher Powers and Fermat's Last Theorem
  • Divisibility and the Greatest Common Denominator
  • Linear Equations and the Greatest Common Denominator
  • Factorization and the Fundamental Theorem of Algebra
  • Congruences
  • Congruences, Powers, and Fermat's Little Theorem
  • Congruences, Powers, and Euler's Formula
  • Euler's Phi Function and the Chinese Remainder Theorem

Unit 2 – Prime Numbers

  • 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

Unit 3 – Quadratic Reciprocity

  • 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?
  • Euler's Phi Function and Sums of Divisors
  • Powers Modulo p and Primitive Roots
  • Primitive Roots and Indices

Unit 4 – Diophantine Equations

  • Square-Triangular Numbers Revisited
  • Pell’s Equation
  • Diophantine Approximation
  • Diophantine Approximation and Pell's Equation
  • The Topsy-Turvy World of Continued Fractions
  • Continued Fractions and Pell's Equation

Unit 5 – Abstract Algebra

  • 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

Unit 6 – Algebraic Number Theory

  • The Equation X4+Y4=Z4
  • 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
  • Generating Functions
  • Sums of Powers

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Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.

This course uses Respondus LockDown Browser proctoring software for designated assessments. LockDown Browser is a client application that is installed to a local computer. Visit the Respondus website for system requirements.

While Chromebook can be used to progress through the course, all exams must be completed on a PC or Mac.