## 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 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.)