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"Combinatorics and Number Theory" by Ken Ono

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.

1st lecture (Ramanujan and partition numbers)

1st talk 125th Ramanujan's birthday

PDF document icon Colloq1_Ramanujan.pdf — PDF document, 1.37 MB (1438850 bytes)

2nd lecture (Congruent numbers and elliptic curves)

2nd lecture on elliptic curves

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

First exercise sheet

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).

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

Some solutions to exercise sheet 1

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!

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

Second exercise sheet

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

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