While waiting for yesterday’s exam to start I read through the unopened
front cover the question “What is a prime?”, and no that wasn’t some
crazy abstract concept but it was actually asking me to write down what
a prime number is. This was swiftly followed by “prove that there are
infinitely many primes”, possibly the most famous proof there is, that
we got in our first week of first year, followed by “state the
fundamental theorem of arithmetic”, again not really degree-level stuff.
There were then a couple more prime number things to prove, both of
which were fairly standard—very frustratingly I couldn’t prove that
there are infinitely many primes of the form 6*k* + 5 despite doing such
a proof by myself before and liking it; I just couldn’t get it to come
out in exam conditions, ah well.

This was yesterday’s paper which was fine, but today’s was disastrous for me. Hopefully with moderating/scaling I should get 50%, but this basically means that “I’ve definitely got 60% average this year” has gone to “hopefully I’ve got 60% average” which is rather unsettling. And today’s paper was supposed to be my best: the first year probability question I was expecting to answer was entirely impenetrable and so I wrote down a few number theory definitions and theorems and did most of a topology question, so that’s not very many marks really. Can’t believe I forgot what sequential compactness is!

It really is bad how Maths tutors are so incapable of writing questions.
It seems that while results get scaled separately by subject, to take
into account the fact that Math/Phils are famously good at Topology and
Maths/Stats people beat everyone else on probability, they are not in
fact, as I thought, scaled by question. On these options papers, there
are questions for different courses and not only are these courses
wildly different in actual difficulty but each question is set by a
different person, that is the course lecturer for that year, so the
chances of getting a consistent difficulty level even within a single
year is pretty low. When you add to this that the lecturer’s conception
of how hard a question is is so *wildly* different to how hard an
undergraduate thinks it is (reflected in examiners’ reports), and you
have a pretty stupid situation. When, like me, you only revise the
minimum number of options, you meet disaster when the questions you want
to answer are too hard to let you do very much at all.

An example of this is that fellow Math/Phil Sophie who had poured her time into the Fields option which I’d dropped a long time ago had a dreadful time yesterday, writing pretty much nothing, whereas today she owned it and came out very happy. We’ve both done similarish amounts of work and have similar levels of ability and interest so that shouldn’t be happening.

I’ve been reflecting on revision and how I’m going to do it better next year. It seems that actually learning everything is probably a good idea, rather than trying to strategise based on past papers, and you’d think that’d be doable next year when we have something like two and a half months of revision. The problem is that the first week of revision is worth less than the last two days, I suspect: proofs don’t stay in long enough. The best I can come up with is: make sure you have understood all the lecture notes as early as possible and have a good set of summary notes inc. sketch proofs, and then spend the middle period doing all the past papers, and then doing them again, and then at the end force bookwork into your head. But then there is too much bookwork. I can’t figure out how to make this work better.

Now it’s all over for the summer and despite the fact that there’s not been anyone around for the better part of a week already, I’ve been left sad now that the year is over. I have to leave Oxford, a place where we say, we’re going to do everything we can, we’re not going to take the easy way out, we’re not going to let up because once one thing finishes, be that academic, social, political or whatever, we’re going to hit the next one equally as hard. It’s exhausting but I fear nothing else will ever be enough. It’s an exhilarating intensity that actually does work, we do come out rather better thinkers. More importantly we’re all that we can be, realistically, on every level. I have to make as much as I can out of the final half of my degree, and out of my time in the unique position of being an Oxbridge undergrad; staying here for graduate study does not mean extending this which you only get three or four years at, and then that has to be it.

*Edit 1/vii/2011:* I’ve since confirmed that the question on Monday that
wasn’t on the syllabus wasn’t, but they won’t do anything, which is fair
enough because some people answered it. The question yesterday that was
impossible to me was impossible because it was on stuff I’d never done
before *i.e.* it was *completely* off the syllabus, not just an
unexaminable *proof*. So carelessness on the part of the examiner has
turned my best paper into my worst `:\`

*The time of this post is inaccurate; forgot to fill it in at the time.*

The 6k+5 thing: off the top of my head, do you do it like this:

Assume there are finitely many primes of this form. List them p1, p2, …, pk. Let N = 6

p1p2*..*pk-1. N mod 6 == -1. Clearly 2, 3 do not divide N so if N has a prime factor it is of form 6k+1 or 6k+5. But if all its prime factors are of form 6k+1, then that would imply N mod 6 == 1. This is not the case: so N has at least one prime factor which is 5 mod 6 (which is the same as -1 mod 6). This can be none of p1, p2, …, pk, and so we have found a prime of form 6k+5 not on the list. Hence assumption of finitely many is incorrect. Hence blah blah. This works?I think that works (your “clearly” bit could be obtained from a previous part of the question).Â Although I am not happy with the jump to the \( 6k+5 \) factor not being any of the \( p_i \)s; I would want to see a little more argumentation there I think, but it’s not hard.

I don’t think any more argument is needed. If q divides N, where N= 6

p1…*pk-1, q cannot be any of the p_i because then q would divide N and N+1, and as q is at least 5 this is patently absurd. It’s fiddling over sticks.