18.781 | Spring 2012 | Undergraduate

Theory of Numbers


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


This course is an elementary introduction to number theory. Topics to be covered include:

  • Primes, Divisibility and the Fundamental Theorem of Arithmetic
  • Greatest Common Divisor (GCD), Euclidean Algorithm
  • Congruences, Chinese Remainder Theorem, Hensel’s Lemma, Primitive Roots
  • Quadratic Residues and Reciprocity
  • Arithmetic Functions, Diophantine Equations, Continued Fractions, etc.


There are no formal prerequisites for the class, but some familiarity with proofs will be helpful as we’ll be doing plenty of those in class and homework. However, if you aren’t used to mathematical proofs, don’t despair! You will hopefully pick up these skills during the course.


There won’t be a required text for the course (we’ll be following lecture notes). There are a few recommended texts, in case you want to do some background reading.

  • Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An Introduction to the Theory of Numbers. Wiley, 1991. ISBN: 9780471625469.

This is quite comprehensive and has a nice collection of topics and exercises. A bit expensive, but if you want to own one book on elementary number theory, this one’s a pretty good candidate.

  • Burton, David M. Elementary Number Theory. Allyn and Bacon, 1976. ISBN: 9780205048144.

This is quite elementary, and explains things in a lot more detail than NZM, so it could be helpful if you haven’t seen proofs before. It’s also pretty comprehensive.

  • Hardy, G.H., and Edward M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1960. ISBN: 9780198533108.

This is a classic.

  • Davenport, Harold, and James H. Davenport. The Higher Arithmetic: An Introduction to the Theory of Numbers. Cambridge University Press, 2008. ISBN: 9780521722360.

It has a very different style from the usual theorem-proof-exercise setup of usual textbooks.

  • Ireland, Kenneth F., and Michael I. Rosen. A Classical Introduction to Modern Number Theory. Springer, 1990. ISBN: 9780387973296. [Preview with Google Books]

Assumes more algebra background, but goes quite far, taking the reader to some of the frontiers of algebraic number theory.

Another source is Franz Lemmermeyer’s lecture notes online (PDF). These assume a bit more background than Niven, Zuckerman and Montgomery, but perhaps not as much as Ireland and Rosen.


There will be a problem set approximately every week, consisting of 5 to 10 problems. The problems are usually not routine calculation, and will require some thinking, so it’s essential to get an early start on the problem set. You are encouraged to collaborate on the homework, but you must write up your own solutions, and I would like you to specify on your homework who was in your working group.

We will also have two in-class midterms and a final exam.


Homework 40%
2 Midterms @ 15% each 30%
Final Exam 30%

Course Info

As Taught In
Spring 2012
Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes