## M3P14: Elementary Number Theory

**Mon 2-4, Tue 3-4 in Huxley 139**

Dr. David Helm

672 Huxley

dhelm@imperial.ac.uk

Office Hours: Mon 4-5, Tue 2-3 (or by appointment)

### 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

Baker, "A concise introduction to the theory of numbers" is a terse, but quite readable book that covers nearly everything
we will discuss in the course. For a somewhat more verbose reference, I'd recommend K. Rosen, Elementary number theory and its applications,
although it suffers from unfortunate "edition bloat"; earlier editions are generally preferable.
### Lecture Notes

For a look at a past year's notes, see
Robert Rouse's 2014 lecture notes here.
I am grateful to Mark Mulcahy for scanning his notes for the early part of the term.

Part 1

Since we have no student notes for the later part of the term, I am compiling an official set of lecture notes.
They are not yet complete but will be posted here as they become available. Note that these have only recently been written and
most likely contain typos; if you're confused by something in the notes please let me know!

Part 1: Euclid's Algorithm and Unique Factorization

Part 2: Congruences and Modular Arithmetic

Part 3: Euler's Theorem

Part 4: Public-Key Cryptography

Part 5: Primitive Roots

Part 6: Quadratic Reciprocity

Part 7: Sums of Two Squares

Part 8: Quadratic Rings and Euclidean Domains

Part 9: Sums of Two Squares Revisited

Part 10: Pell's Equation

Part 11: Continued Fractions

Part 12: Diophantine Approximation

Part 13: Sums of Four Squares

Part 14: Primes in Arithmetic Progressions

### 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. The assignments will be due Tue 25 Oct, Tue 8 Nov, Tue 22 Nov, and Tue 6 Dec.

Example Sheet 1 (Due Tuesday 25 October)
Solutions

Example Sheet 2 (Due Tuesday 8 November)
Solutions

Example Sheet 3 (Due Tuesday 22 November)
Solutions

Example Sheet 4 (Due Tuesday 6 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.)