Number theory

Number theory is the field of mathematics associated with studying the integers.

Introductory Topics

The following topics make a good introduction to number theory.

• Primes
  • Sieve of Eratosthenes
  • Prime factorization
• Composite numbers
• Divisibility
  • Divisors
    • Common divisors
      • Greatest common divisors
    • Counting divisors
  • Multiples
    • Common multiples
      • Least common multiples
• Division Theorem (the Division Algorithm)
• Base numbers
• Diophantine equations
  • Simon's Favorite Factoring Trick
• Modular arithmetic
  • Linear congruence

Intermediate Topics

An intermediate level of study involves many of the topics of introductory number theory, but involves an infusion of mathematical problem solving as well as algebra.

• Diophantine equations
  • Pell equations
  • Simon's Favorite Factoring Trick
• Euclidean algorithm
• Modular arithmetic
  • Linear congruence
    • Chinese Remainder Theorem
  • Euler's Totient Theorem
  • Fermat's Little Theorem
  • Wilson's Theorem

Olympiad Topics

An Olympiad level of study involves familiarity with intermediate topics to a high level, a few new topics, and a highly developed proof writing ability.

• Diophantine equations
  • Pell equations
  • Simon's Favorite Factoring Trick
• Modular arithmetic
  • Linear congruence
    • Chinese Remainder Theorem
  • Euler's Totient Theorem
  • Fermat's Little Theorem
  • Wilson's Theorem
  • Quadratic reciprocity

Resources

• Number Theory by Naoki Sato

Other Topics of Interest

These are other topics that aren't particularly important for competitions and problem solving, but are good to know.

• Fermat's Last Theorem

Famous Unsolved Number Theory Problems

• Birch and Swinnerson-Dyer Conjecture
• Collatz Problem
• Goldbach Conjecture
• Riemann Hypothesis
• Twin Prime Conjecture