The sum is arguably the least mysterious operation in all of mathematics. If you look back to your first experiences in maths, you would probably find that they involved counting and summing. At the same time, studying simple sums of natural numbers can lead to some of the most difficult unsolved problems we have today: Goldbach's conjecture (whether every even number can be written as a sum of two primes) is a famous example of that.

One particularly interesting area which deals with sums of integers is the study of integer partitions. The most general example, unrestricted partitions, i.e. counting the number of ways we can express a given number as a sum of integers (without any sort of restriction on the sum), looks deceptively simple yet gives rise to a surprisingly deep and complex theory. It has been studied by some of the greatest mathematicians, from Euler to Hardy and Ramanujan, using methods ranging from elementary algebra to complex analysis and yields a huge number of fascinating identities and theorems, even when studied with simple combinatorial arguments.

The more classical number theoretic approach deals with proving if certain numbers can be represented as sums of integers from some given set and is no less full of elegant results. An example of this would be Fermat's two squares theorem, which tells us which positive integers can be repesented as sums of two squares and is widely regarded as one of the most beautiful theorems of number theory.

So, if any of this sounds interesting, don't hesitate to come this Thursday at 7:30 PM to MS.04 for a gentle introduction to the awesome world of additive number theory. Then pub.