Dirichlet's Theorem on primes in arithmetic progression states that for any coprime numbers and , we can find infinitely many primes congruent to modulo . A stronger version is in fact true: each residue class contains the same amount of prime numbers: if you write a table of all prime numbers up to and count how many end up in a given residue class (ie how many end up being congruent to modulo ), you will find that there are approximately . This means primes are equidistributed amongst the residue classes, as there are precisely integers between and that are coprime to , and hence only possible infinite families.

Something somewhat similar happens when looking at the splitting of primes in the ring of Gaussian integers : the behaviour is essentially described by Fermat's Theorem on primes expressible as the sum of two squares. Indeed, any prime congruent to mod factors as the product of two prime elements of (for example, ) whereas primes congruent to mod stay prime in this larger ring (lastly, ramifies in this ring: , which means that gains a repeated factor). So here we see that on average, half of the primes split into two factors, and half stay prime: this is related to the fact that the degree , as a Galois field extension, is 2.

This last observation actually generalises much further, and in fact this generalisation was formulated by Frobenius, who was unable to prove it. The Russian mathematician Chebotarev, however, was able to come up with a proof; this generalisation also gives us Dirichlet's Theorem as a special case. The method of proof, that Chebotarev allegedly thought of while carrying water from the lower part of his town to the higher part, has been invaluable in the development of class field theory. Indeed, when Artin first formulated his acclaimed general reciprocity law, he was unable to prove it; only later, after reading Chebotarev's proof, was he able to come up with a proof of his theorem.

Come to MS.05 to learn about all this, and more! After which we factor into the pub.

Just a short note to mention that today's Discussion Group is indeed happening; Ben Simpson will present us with his findings about category theory. Don't miss it!

A short notice: there is a discussion group tonight by Cosmin about some topics in the combinatorics of subdivision, for example on the subject of subdividing a square into even or odd numbers of congruent triangles. But it's going to be good! See you there!

The Warwick Mathematics Society is hosting a special event - a maths Knowitalls contest!

If you're familiar with the BBC2 show Knowitalls, this is the same idea. In the different rounds, each contestant is given a mathematical topic and he has a small amount of time to say everything he knows about the topic.

In the first round, the idea is to come up with as much information as possible, and to mention some of the key points that the jury will have prepared in order to get some sweet bonus points.

In the second round, you will need to come up with one of the key points as fast as possible.

In the third round, you have to come up with as many examples as possible of a given type.

Everyone is more than welcome; we try to make things fair by giving harder topics to third/fourth years than to first or second years. The idea is simply to have fun by trying to remember some nice mathematical facts in familiar areas, not to embarass anyone. But giving impossibly hard questions to Cosmin is also always a fun thing to do; you'll find he doesn't know much about differential geometers after Riemann, for example. Also don't worry if you don't know anything about fiber bundles, we're reserving all those questions for one particular person.

We had a test run with a few regulars last Monday and it proved tremendously fun - as long as you take it lightly, you'll be sure to amuse yourself too! Please be sure to come!