Math 248: Theory of Numbers

Course Information

Syllabus

Official description in the NYU Catalog

Class meets: MW 3:30-4:45pm at 7 East 12th Street, room 125

Instructor: Benjamin Bakker

email: (last name)-at-cims-dot-nyu-dot-edu

Office Hours: W 5-7pm in Warren Weaver Hall 619

Recitations: F 8-9:15am in Warren Weaver Hall 317, starting 9/14

Teaching Assistant: Arjun Krishnan

Homework

• • Homework 0 (due September 12). For this Homework please consult Hatcher's notes. Solutions.

  • Homework 1 (due September 19). Solutions. Mathematica notebook.

  • Homework 2 (due September 26). Please carefully read through my notes. Solutions. Mathematica notebook.

  • Homework 3 (EXTENDED: due October 8). Solutions. Mathematica notebook.

  • Homework 4 (due October 10). Solutions (corrected). Mathematica notebook.

  • Midterm I. Solutions.

  • Homework 5 (due October 29). Solutions. Mathematica notebook.

  • Homework 6 (due November 12). Solutions.

  • Homework 7 (due November 26). Solutions.

  • Quiz I. Solutions.

  • Homework 8 (due December 3). Solutions.

  • Homework 9 (due December 12). Solutions.

  • Quiz II.

  • Final Homework.

  • Final. Solutions.

Lecture Topics and References

Lecture 1: Pythagorean triples, cf. Hatcher's notes. Lecture Notes.

Lecture 2: Properties of the integers and divisibility, cf. Chapter 1 and Appendix A of J&J. For more on the basic axioms satisfied by the integers, see Harron's notes. Lecture Notes.

Lecture 3: The Euclidean algorithm and linear Diophantine equations, cf. Chapter 1 of J&J. Lecture Notes.

Lecture 4: The fundamental theorem of arithmetic and some classical primes, cf. Chapter 2 of J&J. Lecture Notes.

Lecture 5: Factorization in rings and Euclidean domains, cf. my notes. Lecture Notes.

Lecture 6: The ring Z/n, cf. Chapter 3 of J&J. Lecture Notes.

Lecture 7: Prime modulus congruences and Z/p, cf. sections 3.1, 3.2 and 4.1 of J&J. Lecture Notes.

Lecture 8: Composite modulus congruences and the Chinese remainder theorem, cf. Chapter 3 of J&J. Lecture Notes.

Lecture 9: Prime power congruences, cf. Chapter 4 of J&J. Lecture Notes

Lecture 10: Elementary Cryptography. Lecture Notes.

Lecture 11: MIDTERM, covering material up through lecture 8, and the basic idea of lecture 9. Practice Problems.

Lecture 12: The Euler Phi function, cf. Chapter 5 of J&J. Lecture Notes.

Lecture 13: The unit group of Z/p, cf. Sections 6.1 and 6.2 of J&J. Lecture Notes.

Lecture 14: Primitive roots and the equation x^d=a mod n, cf. Sections 6.3-6.5 of J&J. Lecture Notes.

LECTURE CANCELLED 10/29 and 10/31.

Lecture 15: Arithmetic functions and Mobius inversion, cf. Chapter 8 of J&J.

Lecture 16: The Dirichlet product, cf. Chapter 8 of J&J. Lecture Notes.

Lecture 17: Quadratic residues, cf. Chapter 7 of J&J.

Lecture 18: Quadratic reciprocity I, cf. Chapter 7 of J&J. Lecture Notes.

Lecture 19: QUIZ on primitive roots and arithmetic functions. Notes on primitive roots. Practice Problems. Solutions.

LECTURE CANCELLED 11/21.

Lecture 20: Review: Quadratic residues, cf. Chapter 7 of J&J. Lecture Notes.

Lecture 21: Binary quadratic forms. Lecture notes. Lecture Notes.

Lecture 22: Classification by reduced forms. Lecture Notes.

Lecture 23: Representation by forms.

Lecture 24: The Class group.

Lecture 25: Pell's equation.