Number Theory (MMA 300) - HT14

Contact Information




Week-by-week Schedule


List of Examinable Proofs


Old Exams

Old Course Pages

   Contact Information

Lecturer :   Peter Hegarty, Rum MV:L3032, Tel.: (031) 7725371,


I will write my own lecture notes and homeworks, and hand out photocopied extracts from various texts as needed, so there is no required course literature. There are many good number theory books out there, in case you want to have a text of your own. Here are some possibilities :

(NZM) I. Niven, H. Zuckerman and H. Montgomery, An introduction to the theory of numbers (5th edition), Wiley 1991.

There might be some copies of this book in Cremona.

(HR) : G.H. Hardy, An introduction to the theory of numbers.

A new 2008 edition is in the library.

(N) : M.B. Nathanson, Elementary methods in number theory, Springer GTM Series.

This is not in the library.


There will be 23 lectures and 3 homeworks . Examination will be by means of a written exam at the end of the course. The exam will be graded out of 100 points. A mxaimum of 15 bonus points can be obtained from the homeworks (N.B.: these are NOT obligatory !). You will need a minimum of 50 points in total to pass the course.

   Week-by-week Schedule

As we proceed, completed material will be marked in green. The files of lecture notes, together with the material handed out in class, contain the detailed content of the course. Initially, I will work from my notes from the previous edition of the course in 2012. Some changes are anticipated, in particular in the latter part of the course dealing with additive and combinatorial number theory. Any editions will be made in real time, and be available no more than a day after the corresponding lecture.

OBS! The following schedule is approximate and will be continuously updated.

Week Stuff Lecture Notes
45 The origins of number theory in Euclid's Elements (Fundamental Theorem of Arithmetic and the Infinitude of Primes).

Complexity of algorithms (Euclid's algorithm and integer factorisation).

46 Linear Diophantine equations and Frobenius numbers.

First applications of FTA to non-linear Diophantine equations : Pythagorean triples and Fermat's Theorem.

First comments on the distribution of the primes.

47 Estimates for pi(x) from Euclid to Euler.

The Prime Number Theorem (PNT). The Riemann zeta function and heuristic arguments for PNT.

Chebyshev's theorem.

Primes in arithmetic progressions : Dirichlet's theorem.

Back to algebra : the ring Z/nZ (Chinese Remainder Theorem) and the group (Z/nZ)*.

Euler's phi-function. The Fermat/Euler theorem and primality testing.



48 Squares (mod 4) and applications : (i) Primes = 1 (mod 4) (ii) Fermat's theorem on the sums of two squares.

Sums of squares and other classical problems in additive number theory.

Quadratic residues in general : Euler's criterion, Gauss lemma and Quadratic reciprocity.

Quadratic forms.



49 Dirichlet L-functions.

Lagrange's theorem on sums of 4 squares.

Introduction to general additive number theory : sumsets.



50 Bases in general : Sidon sets, thin bases. Combinatorial and probabilistic number theory. PDF

Supplement 1: Cauchy-Davenport theorem

Supplement 2: Probabilistic method

51 Thin bases (ctd.) : Chernoff's inequality and Erdös theorem.

A modern outlook : structure in dense random sets.

Van der Waerden's theorem.

Szemer\'{e}di Regularity Lemma and Roth's theorem.



2 The lectures previously scheduled for this week have now been moved to previous weeks, so there will be no new material this week.
16/1 Exam. 08:30 - 12:30.  


Homework 1 (due Nov. 24) and solutions, here is Q.6(ii)

Homework 2 (due Dec. 12) and solutions

Homework 3 (due Jan. 7) and solutions, here is Q.11

   List of Examinable Proofs



Exam 16/01/15  
PDF   and solutions PDF

Exam 27/08/15   PDF   and solutions PDF

Exam yy/yy/15   PDF   and solutions PDF

   Old Exams

260813 and solutions

020413 and solutions

191212 and solutions

180811 and solutions

181210 and solutions

201208 and solutions

300807 and solutions

170107 and solutions

170105 and solutions

160403 and solutions

180103 and solutions

220801 and solutions

070401 and solutions

270101 and solutions

   Old Course Pages








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Last modified: Sat Aug 29 16:13:00 CET 2015