Maths

[jack]

Last night, I decided I definitely *was* pining for maths, and came home and looked out my numbers and sets example sheets. I was pleased that my storage of maths notes, while not anal, was labelled enough that I could find it, though I couldn't find the lecture notes.

(Numbers and Sets was a course in the very first term, taught by Doctor Leader, notorious for requiring no prerequisites, but being a good introduction to how to think about pure maths, and the example sheet questions graduating from "show you can use the theorems taught" to "no-one solves this ever, apart from a few of the supervisors and professors".)

I've gone through the first half. I could do everything fairly easily that I did fairly easily seven years ago.

Amazingly, I can remember the relevant theorems from the first part of the course after all this time and even how to prove them -- no doubt being the first things, and the first interesting things, they sank in more than everything else[1].

I admit, solving the questions would be basically impossible without remembering them -- you'd basically have to deduce them from the whole cloth, which is possible, but requiring a great leap of intuition. This is something to remember for later courses: often becoming lost was due to not grokking that Blah Theorem was the fundamental part of this section of the course.

My logic is, I need some recreational puzzles, and these are perfect for it: based on knowledge I have, and designed to be solvable by a bright person. And that you can work towards solving, requiring a mix of knowledge, perseverance and insight (well, ok, not much so far). Neither requiring application of only a limited set of skills, nor just the right leap of insight (as eg. soduku or riddles do).

If anyone wants to join me, feel free; I can post the questions and the solutions I do work out :) (I would expect someone bright and interested in maths but without a maths background to be able to make a start, though people may disagree in either direction.)

[1] There are only three: (1) That 1,2...(p-1) all have inverses mod p (2) Fermat's Little Theorem, a^p-1=1 mod p and (3) Wilson's Theorem, (p-1)!=(p-1) mod p.

Comments

[satanicsocks.livejournal.com]

Bizarrely, I have just got back home from Borders, where Dr Leader was two places in front of me in the queue.

[cartesiandaemon.livejournal.com]

Oh, cool. I would have liked to see him; or you.

My invitation to graduation dinner said we should say if we wanted to see if our old tutor or dos was able to come and see us, but it didn't seem worth it, since we're both around and I can just drop into the CMS to say hi. Though if I'm doing N&S example sheets again I *may* end up talking to him...

BTW, do you know about the picnic tomorrow? I thought I couldn't come, but my parents may not be able to be here, in which case I would like to see you all. But Risa's email didn't actually say when or where. Is there a plan, or should I ring someone tomorrow? I have the mobile telephone numbers you had four years ago, but don't know if they are still current.

[satanicsocks.livejournal.com]

Hope this isnt too late, I went to go read HP7 before you replied ;)

It's at 6.30 at Ashleigh's. I don't think the phone number I have for you is up to date either, heh. Mine ends in 037...

[cartesiandaemon.livejournal.com]

:) Thank you! Don't worry, Jenni put me in touch with Ash.

[drswirly.livejournal.com]

I've supervised Numbers&Sets for nearly ten years now, so if you do get stuck then you can always ask for hints. It's by far the loveliest course I do.

Imre has put some new questions on his sheets in the past two years. Here are a couple of the prettiest.

Let a and b be distinct positive integers, with say a < b. Prove that every block of b consecutive positive integers contains two distinct numbers whose product is a multiple of ab. If a, b and c are distinct positive integers, with say a < b < c, must every block of c consecutive positive integers contain three distinct numbers whose product is a multiple of abc?

Is there a positive integer n for which n⁷-77 is a Fibonacci number?

(The first one doesn't require any particular knowledge from the course. The second is harder and does, sort of.)

[cartesiandaemon.livejournal.com]

Thank you! That's very much appreciated. I wondered if you might say something like that, but supervising n&s for work and then coming home and supervising n&s on LJ seemed like a little much :) I'll let everyone know how I do; I expect questions near the end will be hard enough I need advice or confirmation, but easy enough I don't give up entirely :)

Imre has put some new questions on his sheets in the past two years. Here are a couple of the prettiest.

Ooh, thank you. And of course I remember the "Your series certainly wins the `brilliancy prize' for Q14!" question you were thinking about before :)

[drswirly.livejournal.com]

supervising n&s for work and then coming home and supervising n&s on LJ seemed like a little much

It's Numbers&Sets! How could anyone get bored by it?

Besides, it's been two months since I taught any, and it's two months until I do so again. Perhaps I'm suffering from withdrawal symptoms.

[cartesiandaemon.livejournal.com]

LOL. I guess that's what *I* think, though most people I know DID get bored by numbers and sets; and as often as not switched to physics or compsci :)

Well, I'll let you know. So far I'm nearly word-for-word the same as what I wrote seven years ago, so I'm probably right. (Though sometimes I just naturally go down the same path :))

[alextfish.livejournal.com]

Oh, I miss maths as well. I'd like to see some of these, just to see how I do. I can at least prove in my head your theorems (1) and (2) and solve the first of drswirly's questions from his comment.

[alextfish.livejournal.com]

I think I've proved Wilson's Theorem too, although I certainly think it was newly duplicated reasoning rather than remembered reasoning.

[cartesiandaemon.livejournal.com]

I'm glad someone else is interested :) I found I could remember the key tricks, and managed to put them back into the proofs -- compared to the rest of the course, they weren't as hard as I remember, though I don't know if I could ever have got it from scratch :)

(I would have thought it necessary to write it down (unless you're familiar with it) -- I find this sort of thing extremely prone to implicit assumptions, especially as going back to the beginning. I expect your proofs are right, though I admit doubt they're rigorous if they're not written down. But maybe that's just me.)

FWIW, you can probably find the proofs online, but what I have is:

(1) Have 1<a<p. Consider a,2a,3a,...,(p-1)a.

These are non-zero mod p, else p divides ai. But p is prime, so must divide one or the other (proof by fundamental theorem of arithmetic, that each number is a unique product of primes), contradiction, as both <p.

Also these must be distinct else some ia=ja (1<=j<i<p) hence (i-j)a==0 (p) hence p divides (i-j)a. Contradiction as before.

But then a,2a...(p-1)a must be equal (mod p) 1,2...,p-1 in some order, hence some ai==1 (p), so i is the required inverse.

(2) (a)(2a)...((p-1)a)==(1)(2)...(p-1) as both lists are the same rearranged (see before). Then a^p-1(p-1)! == (p-1)! (mod p). (p-1)!=/=0 since no product of a,i<p can be a multiple of p. So cancel it from both sides and a^p-1==1 (p)

(3) What are self-inverses? a^2==1 (p) <==> a^2-1==0 <==> (a+1)(a-1)==0 <==> a+1=p (since 1<=a<p, and p| a+1 or a-1 ). So p-1 is the only self-inverse.

Hence (p-1)! is a product of: 1, p-1, and (p-3)/2 pairs of inverses. All the inverses cancel, leaving (p-1)!==(p-1) mod p.

Does that accord?

[alextfish.livejournal.com]

I sat down to write them out before looking at this comment, and you were right, I'd neglected to prove that ia=/=ja mod p for 1<=j<i<p until that point. And I'd also handwaved rather about the self-inverses. But otherwise, yes, that lines up with what was in my head. Yay, maths :)

[cartesiandaemon.livejournal.com]

:) I'm glad I wasn't maligning you too much. I was exactly the same at the time, I always saw the answer and never liked clearing up the edges, however necessary I recognised it. It's obviously not important in this case -- if you needed to be rigorous, you could prove that as soon as you thought of it.

It's a good feeling to know the skill is all still there after all, isn't it? :)

Yay, maths :)

:) Likewise. I've been puzzling all day at work, the first, um, six-ish questions I've been doing were fairly easy, but the next I'm blocked on. I need to check my notes -- there may be another result I'd forgotten which is necessary. Else I just haven't thought how to do it yet.

[cartesiandaemon.livejournal.com]

Did you study maths? I can certainly provide the questions, though drswirly may know if they're online somewhere already.

[alextfish.livejournal.com]

Yes, I did 4 years of pure maths. This is probably why I miss it. I'd have liked to do a PhD in Category Theory or some kind of set theory or provability, but didn't get the grade in Part III.

[cartesiandaemon.livejournal.com]

Ah! I may have known that, but I didn't remember for sure. I was in much the same position (except for category theory -- it was definitely a cool idea, but I could never get into it :( ).

I would have guessed you were a pure mathmo if anything, but don't like to assume -- some compscis get offended :)

Well, I'd appreciate the company, for all the same reasons I missed maths myself. I decided I probably didn't want to ever go back and do a PhD, but maths for fun was probably a good idea, since I missed the challenge. And if I wanted to do a PhD studying serious maths myself would probably be equivalent, though I doubt I'll do that.