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Number theory · Three-course sequence

Number Theory

A three-course, proof-based sequence in number theory, from divisibility and congruences to quadratic reciprocity, Dirichlet characters, and L-series. Each course builds on the one before.

Introduction to Number Theory

Introduction to Number Theory is a proof-based course introducing the fundamental ideas and methods of elementary number theory through problem solving and rigorous mathematical reasoning. Topics include divisibility, prime numbers, greatest common divisors, modular arithmetic, classical Diophantine equations, Pell equations, continued fractions, Fermat’s Little Theorem, Euler’s Theorem, and applications of congruences.

  • Divisibility
  • Primes
  • GCDs
  • Modular arithmetic
  • Diophantine equations
  • Pell equations
  • Continued fractions
  • Fermat & Euler

Intermediate Number Theory

Intermediate Number Theory builds on the PiMath Introduction to Number Theory course, developing deeper structural results through rigorous problem solving and mathematical exposition. Topics include Mersenne primes, perfect numbers, order and primitive roots, Diophantine equations, Fibonacci numbers, continued fractions, Pell’s equations, geometric number theory, and Farey sequences.

  • Mersenne primes
  • Perfect numbers
  • Primitive roots
  • Fibonacci numbers
  • Continued fractions
  • Pell’s equations
  • Geometric number theory
  • Farey sequences

Advanced Number Theory

Advanced Number Theory is a continuation of Introduction to Number Theory, emphasizing deeper theoretical results, synthesis across topics, and sustained work on advanced and olympiad-level problems. Topics include quadratic residues, quadratic reciprocity, roots of unity, congruent numbers, the BSD conjecture, primes in arithmetic progression, Dirichlet characters, Gauss and Jacobi sums, and Dirichlet L-series.

  • Quadratic reciprocity
  • Roots of unity
  • Congruent numbers
  • BSD conjecture
  • Dirichlet characters
  • Gauss & Jacobi sums
  • Dirichlet L-series

Every PiMath course

How classes work

Live online

Seminar-style discussion over Zoom. Every session is recorded.

12 students at most

Small enough that every student is heard and known.

Canvas course site

Schedule, recordings, annotated slides, and course notes in one place.

Feedback on every proof

Detailed comments on problem solving and proof writing.

Ready to apply?

Placement is based on readiness and interest rather than age or grade level. If you are not sure which course fits, choose “Not sure which course fits — please advise” on the application and we will help.

Open the application formQuestions? Contact us

Financial aid and scholarships

Partial and full need-based scholarships are available for every course. Ask about aid on the application.

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