Introduction to Classical Number Theory

Introduction to Classical 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 will include divisibility, prime numbers, greatest common divisors, modular arithmetic, classical Diophantine equations, Pell equations, continued fractions, Fermat’s Little Theorem, Euler’s Theorem, and introductory applications of congruences. Emphasis is placed on precise proof writing, sustained engagement with challenging problems, the development of mathematical maturity, and clear mathematical exposition. This course is particularly well suited for students who are new to mathematical proof, transitioning into advanced proof-based mathematics, or seeking to strengthen their mathematical reasoning and problem-solving skills in a rigorous and supportive environment. Although the focus of this course is not on contest preparation, we will work on several problems at the level of the AMC 10, AMC 12, AIME, and USAJMO exams.

Advanced Number Theory: Topics in Algebraic and Analytic 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 may include quadratic residues, quadratic reciprocity, arithmetic functions, continued fractions, Mobius inversion, Dirichlet convolution, Dirichlet series, L functions, and other topics in algebraic number theory and analytic number theory. The course places strong emphasis on elegant proof techniques, conceptual connections across number theory, Olympiad-level problem solving, mathematical exposition, and preparation for independent inquiry and research in number theory and related areas. Students enrolling in this course are expected to have prior experience with rigorous mathematical proof and familiarity with the core topics of an introductory proof-based number theory course, including divisibility, modular arithmetic, congruences, Diophantine equations, and foundational results such as Fermat’s Little Theorem and Euler’s Theorem. Although the focus of this course is not on contest preparation, we will work on several problems at the level of the USAMO and IMO exams.