Combinatorics and Number Theory are old mathematical subjects which are home to many simple to state open problems. This course will be a brief introduction to classical problems in combinatorics and number theory which quickly lead to the theory of modular forms. We shall discuss classical problems involving right triangles, partitions of integers, and the distribution of primes. The selected problems will highlight some of the deeper phenomenon which underneath seemingly innocent questions.

2nd lecture on elliptic curves

Colloq2_EC.pdf
—
PDF document,
835 KB (855377 bytes)

An exercise sheet on partition numbers, where among other things you get to prove the following theorem of Rammanujan:
5 divides every p(5n+4).

hw1.pdf
—
PDF document,
36 KB (36974 bytes)

Here are sketches of the answers to the first two exercises of the first discussion session for Ken Ono (on partition numbers). If these are not precise enough, complain!

Ono1solutions.pdf
—
PDF document,
130 KB (133294 bytes)

This exercise sheet deals with congruent numbers, and elliptic curves.

hw2.pdf
—
PDF document,
38 KB (39475 bytes)