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

Introductory Resources

Books

• The AoPS Introduction to Number Theory by Mathew Crawford.

Classes

• Introduction to Number Theory Course Details

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

Intermediate Resources

Classes

• Intermediate Number Theory (Details)

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

Olympiad Resources

Books

• Number Theory by Naoki Sato

Advanced Topics in Number Theory

Algebraic Number Theory

Algebraic number theory studies number theory from the perspective of abstract algebra. In particular, heavy use is made of ring theory and Galois theory. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the study of . Famous problems in algebraic number theory include the Birch and Swinnerson-Dyer Conjecture and Fermat's Last Theorem.

Analytic Number Theory

Analytic number theory studies number theory from the perspective of calculus, and in particular real analysis and complex analysis. The techniques of analysis and calculus are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem in analytic number theory is the Riemann Hypothesis.

Elliptic Curves and Modular Forms

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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

Books

Here are a list of general interest books:

• Fermat's Enigma by Simon Singh (details)
• Music of the Primes by Marcus du Sautoy (details)