Math 304 — Spring 2025
This course offers an introduction to elementary number theory with minimal background prerequisites. Following George Andrews’ Number Theory, we will cover essential concepts and some of the most celebrated results in elementary number theory, divisibility, the theorems of Fermat, Euler, and Wilson, the Chinese remainder theorem, prime numbers and factorization, some arithmetic functions, primitive roots, quadratic reciprocity, and Diophantine equations. Time permitting, additional topics from later chapters in the book, such as Pell’s equation, continued fractions, or factorization in the Gaussian integers, may also be included.
We will roughly follow the book Number Theory by George Andrews, and course notes provided by the lecturer.
The first midterm covers content up to February 13th.
The second midterm covers content up to the spring break.
chapter:inductionchapter:euclidean-division-and-representabilitychapter:linear-diophantine-equationschapter:fundamental-theorem-of-arithmeticchapter:combinatorial-methodsfrom these notes, and Chapters 1, 2, and 3 from Andrews, Number Theory.
Note: parts of Chapter 3 of Andrews’ book do not have an equivalent in these notes, so don’t forget to study Chapter 3 of Andrews.
chapter:polynomialschapter:congruenceschapter:modular-arithmeticchapter:systems-of-linear-congruenceschapter:polynomial-congruenceschapter:primitive-rootsfrom these notes, and Chapters 4, 5, 6.1, and 7 from Andrews, Number Theory.
The course grade will be determined as follows:
This adds up to a total of 120%, allowing students the opportunity to earn extra credit.
The grading scale is:
If the median score for the class falls below a B, grades may be adjusted (curved) so that the median corresponds to a B. Any adjustments will be made consistently for all students.
First class! We give a short history of number theory, and get started with mathematical induction.
We review mathematical induction and prove the well-ordering theorem. We will then use the well-ordering theorem to prove the Euclidean division theorem and the representability theorem.