Proof-based mathematics · Live online · Winter 2027 courses announced
PiMath offers rigorous, proof-based online math courses and research mentorship for advanced high school and middle school students who want to go beyond the traditional school curriculum. Small live seminars, real proofs, and individual mentorship from a Stanford Ph.D. mathematician.
January – March 2027
Winter 2027 courses, plus research and individual mentorship. Courses build on earlier PiMath coursework or equivalent preparation.
Abstract algebra
This course introduces the algebraic structure of rings, exploring how familiar properties of the integers and polynomials extend to more general settings. Topics include ideals, quotient rings, ring homomorphisms, integral domains, polynomial rings, and unique factorization. Students develop a deeper understanding of algebraic structures through rigorous proofs and problem solving.
Builds on: Group Theory
Tuition$1,200
Full course details →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.
Builds on: Introduction to Number Theory
Tuition$1,000
Full course details →Contest preparation
This course prepares students for the American Invitational Mathematics Examination, the next step after the AMC 10 and AMC 12. Students develop the deeper techniques that AIME problems require in algebra, combinatorics, geometry, and number theory, working through past AIME problems and presenting their solutions to the class.
Dates: December 8, 2026 through the AIME (Feb. 5–6, 2027)
Schedule: Sundays and Tuesdays, 4:00–5:00 pm PT
Builds on: AMC Preparation or strong AMC 10/12 experience
Tuition$800
Full course details →Research and one-on-one
During the academic year, the Research Seminar is offered as individual mentorship. Students work one-on-one with the instructor to explore advanced topics, develop original results or meaningful extensions of known theorems, and learn the practices of mathematical research. Mentorship can also be built around other goals, such as studying an advanced topic, strengthening proof writing, or preparing for competitions. Each plan is designed together with the student and family.
Builds on: prior proof-based coursework
TuitionDetermined individually for each student
Full course details →Our approach
PiMath courses draw on the traditions of math circles, advanced problem-solving seminars, and research-style inquiry. Every course rests on three commitments.
Students learn to write complete, rigorous arguments. Proof-based coursework builds advanced reasoning and the mathematical maturity that university mathematics requires.
Students experiment, notice patterns, make conjectures, and test them. Collaborative discussion turns hard problems into shared investigations.
Every student receives detailed, individual feedback on problem solving and proof writing, with guidance that continues from course to course.
We do not race through the standard sequence. We explore why mathematics works, not only how to apply formulas.
Students learn to build arguments and communicate them with precision.
Many problems rival olympiad questions in depth, but the focus is long-term mathematical growth.
Students think deeply about ideas over time, with opportunities to continue from course to course.
Courses
Courses are organized in tracks, so students can continue from one term to the next.
Foundations
This course is designed for students who are curious about mathematics beyond the standard curriculum and eager to explore ideas in greater depth. The course emphasizes number theory, number systems, and geometry, with a strong focus on reasoning, pattern-finding, and clear mathematical explanation.
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.
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.
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.
Abstract algebra
Introduction to Group Theory is a proof-based course exploring symmetry and algebraic structure through rigorous problem solving and mathematical exposition. Topics include groups and subgroups, symmetric and dihedral groups, cyclic groups and orders of elements, cosets and Lagrange’s theorem, homomorphisms and isomorphisms, quotient groups, group actions, Burnside’s lemma, and the Sylow theorems.
Abstract algebra
This course introduces the algebraic structure of rings, exploring how familiar properties of the integers and polynomials extend to more general settings. Topics include ideals, quotient rings, ring homomorphisms, integral domains, polynomial rings, and unique factorization. Students develop a deeper understanding of algebraic structures through rigorous proofs and problem solving.
Abstract algebra
This course explores the structure of fields and their extensions, with an emphasis on understanding polynomial equations and their solutions. Topics include field extensions, algebraic and transcendental elements, minimal polynomials, splitting fields, and finite fields. Students investigate how algebraic structures provide a framework for studying roots of polynomials and connections between different number systems.
Abstract algebra
This course develops the beautiful correspondence between field extensions and groups of symmetries, culminating in the Fundamental Theorem of Galois Theory. Topics include field automorphisms, splitting fields, Galois groups, and solvability of polynomial equations by radicals. Students explore applications to classical geometric constructions and the impossibility of a general formula by radicals for polynomial equations of degree five or higher.
Problem solving
Advanced Problem Solving & Proof is a rigorous, proof-based mathematics course designed to help students learn how mathematicians think, reason, and communicate. The course emphasizes deep problem solving, logical reasoning, and clear mathematical writing rather than speed or rote techniques.
Contest preparation
This course prepares students for the AMC 10 and AMC 12 competitions through a focused study of the techniques that appear most often on these exams. Rather than drilling problems in isolation, we develop the underlying ideas and then see how they play out across many contests, so that students leave with transferable methods rather than memorized tricks. Students preparing for the AMC 8 are also welcome: the course develops exactly the tools needed for the most challenging problems on that exam.
Topics include counting and combinatorial techniques, number theory (divisibility, modular arithmetic, and Diophantine equations), algebraic manipulation and polynomials, sequences and series, plane and coordinate geometry, trigonometry, complex numbers, probability and expected value, and general problem-solving strategies such as invariants, extremal arguments, and clever substitutions. Throughout, students work on problems from past AMC contests, including recent exams, and present their solutions to the class.
Research
An advanced, seminar-style course designed for students who are ready to engage in sustained, independent mathematical inquiry. Building on prior coursework in proof-based mathematics, students work closely with the instructor to explore advanced topics, develop original results or meaningful extensions of known theorems, and learn the practices of mathematical research.
During the academic year, the Research Seminar is offered as individual mentorship.
Course offerings vary by term. Individual mathematics mentorship is also available.
A taste of PiMath
Euler proved this in 1735. In class we prove it more than once, from completely different directions:
Look at the last two rows. Why is the missing amount almost exactly 1/N? And why should a sum of fractions involve π at all?
Student research
In the Research Seminar, students work on problems that are easy to state and still open. They compute examples, look for patterns, make precise conjectures, and prove what they can.
Seminar work has led to new theorems and to papers written with students, including several published papers in peer-reviewed journals.
Theorem (Li–Paquin)
Every lattice triangle with exactly three boundary points and k interior points has its interior points on one line exactly when k = 1, 2, 4, or 7.
Seminar research areas
Papers written with student researchers. Student coauthors are shown in bold.
Minkowski’s theorem on Archimedean lattices
D. Paquin and J. Schennach
The American Mathematical Monthly 133(6), 578–583
Numerical analysis of critical parameter values for remission during imatinib treatment of chronic myelogenous leukemia
D. Paquin, L. Gross, A. Stewart, and G. Thai
Mathematical Biosciences and Engineering 22(6), 1551–1571
Modeling free tumor growth: Discrete, continuum, and hybrid approaches to interpreting cancer development
D. Singh and D. Paquin
Mathematical Biosciences and Engineering 21(7), 6659–6693
A mathematical model for the effects of grandmothering on human longevity
D. Paquin, D. Kato, and P. Kim
Mathematical Biosciences and Engineering 17(4), 3175–3189
An analysis of strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia with imatinib-resistant mutations
D. Paquin, D. Sacco, and J. Shamshoian
Mathematical Biosciences 262, 117–124
On the magic positivity of the Ehrhart polynomials of dilated base polytopes of minimal matroids
J. Chen, C. Crigler, S. Ding, D. Paquin, and J. Schennach
Under review, Annals of Combinatorics
The one-power Erdős conjecture for primes p ≡ 7 (mod 36)
S. Harman and D. Paquin
Under review, Integers
Collinear interior lattice points in triangles satisfying B(T) ∈ {4, 5}
A. Sakunkoo, J. Sakunkoo, and D. Paquin
Submitted to Integers · arXiv:2608.00889
On lattice triangles satisfying B(T) = 3 with collinear interior lattice points
E. Li and D. Paquin
Submitted to Notes on Number Theory and Discrete Mathematics · arXiv:2501.15009
Convex lattice polygons with k ≥ 3 interior points
D. Paquin, E. Sumera, and T. Tran
Submitted to Journal of Geometry · arXiv:2501.18003
On coprime-preserving transformations and dynamic coprime labeling
A. Tonapi and D. Paquin
What a class looks like
Taught over Zoom with seminar-style discussion. Every session is recorded.
Small enough that every student is heard and known.
Schedule, recordings, annotated slides, and course notes in one place.
Regular homework with detailed comments on problem solving and proof writing.
Instructor
Founder and Lead Instructor. More than two decades of teaching mathematics to talented students.
PiMath courses are designed for secondary students who are eager to explore mathematics beyond the traditional school curriculum and who enjoy thinking deeply about challenging problems. These courses emphasize proof, structure, and mathematical reasoning, and are well suited for students who are curious about why mathematics works, not just how to apply formulas.
The program is not focused on contest preparation, though many of the problems encountered are comparable in depth and difficulty to advanced olympiad-style questions. Instead, the emphasis is on long-term mathematical growth, clear written exposition, and sustained engagement with ideas over time.
Highly motivated middle school students are also welcome to apply, particularly those with prior experience in proof-based mathematics or enrichment programs. Our Transition to Advanced Mathematics and Introduction to Number Theory courses are designed specifically for advanced middle school students and early high school students. Placement is based on readiness and interest rather than age or grade level.
PiMath is committed to making rigorous mathematics accessible to motivated students. Partial and full need-based scholarships are available for every course. Ask about aid when you apply.
Apply
Read the course descriptions above and note which tracks interest your student.
Pick the course that fits your student's background. If you are unsure, write and we will help you decide.
Complete the online application, selecting one or more courses. We will be in touch about placement, schedule, and financial aid.