## M3P14: Elementary Number Theory

**Mon 10-11, Tues 11-12, Fri 11-12 in Huxley 139**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

Office Hours: Mon 3-5

### Course description

This course is an introduction to elementary number theory. We will study topics
including factorization, the distribution of primes, and modular arithmetic, as
well as some simple diophantine equations.
### Suggested References

T. Apostol, Introduction to analytic number theory

K. Rosen, Elementary number theory and its applications (this suffers from unfortunate "edition bloat";
earlier editions are generally preferable)

K. Ireland and M. Rosen, A classical introduction to modern number theory (this is somewhat more
advanced than the first two)

### Lecture Notes

I am very grateful to Alina Khaybullina for allowing her lecture notes to be put online. They are now in manageable pieces:

part 1

part 2

part 3

part 4

part 5

part 6

part 7

They are current as of Friday, 11 December. Last year's lectures,
which should be similar (but probably not identical) to this year's, were taken by Robert Rouse, and can be
found
here.
### Problems

Example Sheets will be posted here every two weeks. They will not be assessed work, but they will be
collected, two weeks after being assigned, and marked. This is purely optional if you want feedback
on the assignments.

Example Sheet 1 (Due Monday 2 November)
Solutions

Example Sheet 2 (Due Monday 16 November)
Solutions

Example Sheet 3 (Due Monday 30 November)
Solutions

Example Sheet 4 (Due Monday 14 December)
Solutions

### Mastery Material

The mastery material for fourth-year students in this course is about the ring of p-adic integers
and Hensel's Lemma, and is based on these notes.
These notes contain many exercises; they will not be assessed, but they are essential to understanding
the subject. Most of the exercises are straightforward.
I will be happy to answer questions (within reason) about the material (in particular,
if you think you have found a mistake or typo, please let me know.)