Triangles, parallelograms and Shimura curves: Maths!

work, studying, science, nature
gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Gordon wrote:No, I meant if the digits of x add up to a multiple of 9, then x is a multiple of 9.
Absolutely, yeah - I was thinking out loud when I wrote that last post. It's a nice divisibility test, that one, isn't it?

I do agree with you, Martijn, that the set of numbers that the constant difference c can be is small. What I still think is an open and interesting question though is whether there's some similarity between the prime sequences that have a 3330 difference. That particular difference of 3330 seems to recur no matter how you vary the number of digits in the prime (though I've not gone beyond length-3 sequences as yet, and I should say that other values of c also recur). So I wonder if the particular kind of permuting being done in those sequences is somehow similar (or perhaps the same). If you could identify 'classes' of prime sequence of this sort - the class with difference 3330, for instanc - that could be maybe interesting. Maybe.

The only progress I've made though is to show that for most permutations of abcd the difference c will be given by something like 90[10(a-c) + (b-c)]. So in effect, for some permutations of a 4-digit prime, you construct the 2-digit number 10(a-c) + (b-c) from its digits and that gives you the multiple of 90 that the difference c will be. And in our cases - for 4817->1847 or 6299->2969 - a-c and b-c are the same, and the number 10(a-c) + (b-c) is 37, and 37.90 is 3330.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Big Nose
Posts: 1289
Joined: Tue Oct 02, 2007 9:27 am

Re: Triangles, parallelograms and Shimura curves: Maths!

These are just artifacts in the base10 number system. They are there in any other base you choose to work with. I was attempting earlier, in a particularly lucid 10 mins, to imagine square and triangles and pentagons in a particularly awesome topographical type way. I felt on the edge of a personal mathematical breakthrough, but then something saucy came on tv and I lost it.

9 isn't an especially special number, well in base10 it is. base-1 is an especially special number.
My apple pies go off today.

Martijn
Posts: 1260
Joined: Sat Oct 13, 2007 8:27 pm
Last.fm: http://www.last.fm/user/thinksmall
Location: Exeter
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Martijn wrote:Random maths fact of the day: the only three arithmetic sequences whose members are cyclic permutations of each other and who are of maximal length are the ones starting with 148, 259 and 012345679.
Erm. That's not true. These are the only ones where the digits of the first term are in ascending order, but that's much less interesting. (296-629-962 --thanks, David-- is a counterexample).

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Big Nose wrote:These are just artifacts in the base10 number system. They are there in any other base you choose to work with. I was attempting earlier, in a particularly lucid 10 mins, to imagine square and triangles and pentagons in a particularly awesome topographical type way. I felt on the edge of a personal mathematical breakthrough, but then something saucy came on tv and I lost it.

9 isn't an especially special number, well in base10 it is. base-1 is an especially special number.
Sorry, I mean to reply to your post as well! I was going to ask if the relationship between the underlying things themselves wouldn't be there irrespective of base, but in trying to write the reply I've convinced myself that I'm talking nonsense.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Well, maybe half-nonsense. If hypothetically there were an underlying reason why the primes 2969/6299/9629 and 1487/4187/8147 (and others) shared a common difference, in whichever base, then I think that would be interesting. Although it would be so interesting that it would be ridiculous that no one's looked into it before. But I agree with you that trying to get somewhere with it on the basis of rotations of base-10 digits is pretty arbitrary. And defining the ends of sequences that way is pretty arbitrary too - really we (I) should just look at sequences of primes and forget the permuting thing entirely. In my mind I was thinking of links to permutation groups, which is really very silly, on reflection.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Big Nose
Posts: 1289
Joined: Tue Oct 02, 2007 9:27 am

Re: Triangles, parallelograms and Shimura curves: Maths!

gloom button wrote:
Big Nose wrote:These are just artifacts in the base10 number system. They are there in any other base you choose to work with. I was attempting earlier, in a particularly lucid 10 mins, to imagine square and triangles and pentagons in a particularly awesome topographical type way. I felt on the edge of a personal mathematical breakthrough, but then something saucy came on tv and I lost it.

9 isn't an especially special number, well in base10 it is. base-1 is an especially special number.
Sorry, I mean to reply to your post as well! I was going to ask if the relationship between the underlying things themselves wouldn't be there irrespective of base, but in trying to write the reply I've convinced myself that I'm talking nonsense.
I don't know, I am crap at maths. But I suspect that the base is irrelevant.

In base10 111111111 x 111111111 = 12345678987654321

Beautiful but obvious.

In octal 111111111 x 111111111 = 12345701207654321

Which lacks the symmetry but is equally obvious if you can forget decimal for a while.

I was always poor at maths, but was a software engineer spending most of my time working in base2, base16 or base8, so I can generally think outside the decimal box. The wonders exist in all bases (patterns in binary numbers are a classic example).
My apple pies go off today.

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Something completely unrelated and ace I was reading the other day I meant to post here too. In Russell's book about Western philosophy he talks a little about Greek mathematicians and their approximating of the square root of 2. Sorry if this terribly well known, but I'd never heard it before and I think it's amazing and beautiful and genius.

So they write two columns of numbers, a and b, and start each with 1 and 1. And to get the next a they take the previous (a+b). And to get the next b they take twice the previous a plus the previous b.

So then for every pair (a,b), 2a^2 - b^2 is 1 or -1, so that dividing across by a^2 you get a right-hand side that tends to 0 as a becomes large and a left-hand side with b/a getting closer and closer to the square root of 2. So by the fourth iteration they have 99/70 as their approximation which is accurate to four significant places.

I think it's the most ingenious thing I've ever heard.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

lynsosaurus
Posts: 3469
Joined: Fri Sep 28, 2007 6:13 pm
Last.fm: http://www.last.fm/user/
Location: Auld Reekie
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

i wish i was as smart as you guys. geography seems so.... clumsy, compared to maths.

Gordon
Posts: 5340
Joined: Tue Feb 05, 2008 10:33 pm
Last.fm: http://www.last.fm/user/GreenGordon
Location: King's Landing
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

lynsosaurus wrote:i wish i was as smart as you guys. geography seems so.... clumsy, compared to maths.
Pfft, my degree subject is 'reading the news'.
Toot toot.

Big Nose
Posts: 1289
Joined: Tue Oct 02, 2007 9:27 am

Re: Triangles, parallelograms and Shimura curves: Maths!

My apple pies go off today.

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Actually my brother sent me that in the course of doing one of the earlier problems we were doing. The primes section is nuts isn't it? They look for all the world like iron filings on a sheet with a magnet underneath it. I like how the roots of those crazy polynomials like x^2 - x - 41 come up in such pretty patterns.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Martijn
Posts: 1260
Joined: Sat Oct 13, 2007 8:27 pm
Last.fm: http://www.last.fm/user/thinksmall
Location: Exeter
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Big Nose wrote:Oof! This is madness.

http://www.numberspiral.com/index.html
Hey, that's dead exciting. Thanks for posting that! It is fascinating to see how certain 'curves' appear to contain more primes than others, although I'm sure this will be equalled out if you make the spiral bigger. (I think all known results about the distribution of primes say that they are, indeed, randomly distributed. It might say this on that site; I have yet to read the final few pages.)

soft revolution
Posts: 2058
Joined: Tue Oct 02, 2007 6:48 pm
Last.fm: http://www.last.fm/user/duncydunc
Location: somewhere in the bullring
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

I've got a stats question if anyone can help?

If you've got a series of monthly figures which decreases in the summer and increases in the winter (say for example cases of flu or something), what test can you use to see if it's a statistically significant change?

I've worked out the mean and standard deviation, but that's as far as my knowledge gets me.

Edit (just worked it out - by calculating the Z and using a standard normal distribution table)
And by me, I mean, Flexo.

frogblast
Posts: 324
Joined: Tue Oct 02, 2007 7:18 pm
Last.fm: http://www.last.fm/user/frogblast
Location: Edinburgh

Re: Triangles, parallelograms and Shimura curves: Maths!

all statistics is a lie.

well, not really, but i think it's a nice example of the usefulness of graphs and the influence of outliers, and i was abnormally excited by it the first time i saw it.

the 4 datasets in these graphs all have the same means, variances, correlations and linear regression lines
Sun like honey on the floor,
Warm as the steps by our back door.

Martijn
Posts: 1260
Joined: Sat Oct 13, 2007 8:27 pm
Last.fm: http://www.last.fm/user/thinksmall
Location: Exeter
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

I am helping a befriended maths professor by typesetting his notes for a course he is due to give (entering it into LaTex, more precisely; he's paying me well for it). It is in modular forms and modular curves. And I just found myself entering the Taniyama-Shimura conjecture, the proof of which meant the solution of Fermat's Last Theorem, arguably the most-celebrated mathematical achievement of the last decades.

I'm not entirely sure if the proof itself is given in this lecture, even less so if I will understand it, but still, it's quite cool. (But only if you're a mathematician, I am afraid.)

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Martijn wrote:
Big Nose wrote:Oof! This is madness.

http://www.numberspiral.com/index.html
Hey, that's dead exciting. Thanks for posting that! It is fascinating to see how certain 'curves' appear to contain more primes than others, although I'm sure this will be equalled out if you make the spiral bigger.
My brother and me are doing another Project Euler problem at the moment involving those number spirals. It's quite a nice one - http://projecteuler.net/index.php?secti ... lems&id=58 - about why the 'corners' of a spiral initially contain surprisingly many primes. Well, it's not about the why so much as figuring out how the distribution levels off, but understanding the why (which comes down to looking at polynomials) helps in writing a smarter program to answer the question. Well, I think: we haven't quite got it yet.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Martijn
Posts: 1260
Joined: Sat Oct 13, 2007 8:27 pm
Last.fm: http://www.last.fm/user/thinksmall
Location: Exeter
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

This is nice stuff to think about while waiting for a train. :-)

So after a lot of complicated calculations, I concluded that I didn't need all those complication to find the polynomials.

I'm not sure if I agree with the "surprisingly many" bit though. They're all odd numbers. 14 of the 25 odd numbers up to 49 are prime, which is 56%, so 62% is hardly surprising.

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Well, quite. I should watch my adverbs. Slightly surprisingly many, maybe.

Actually, my terribly clever idea turns out, I think, to be terribly stupid. I was trying to think of things to make the programming part simple, and thought that, instead of dealing with three new numbers on the diagonals at each iteration of the spiral, I could think of them as an array comprising the square number on the lower right diagonal offset by a (predictable) constant. So at the fourth iteration, for example, you'd have 25 with the other three diagonal elements given by [25-4, 25-8, 25-12] respectively.

And then I was thinking that that actually itself gives quite a nice account of why 21 is the first non-prime on a diagonal, being just a (factorisable) difference of two squares. I thought that would generalise quite nicely into a solution, but there's other tricksiness 'cause while it's easy to predict that a difference of squares isn't prime, it's harder to predict if some differences (like 11^2 - 10 = 91 = 3*17) will be factorisable. And I think that's getting into diophantine equations (unless my memory of what a diophantine is is failing).

It's quite depressing, actually, not being able to figure this out. I'm sure I'm supposed to be better at things than this.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Martijn
Posts: 1260
Joined: Sat Oct 13, 2007 8:27 pm
Last.fm: http://www.last.fm/user/thinksmall
Location: Exeter
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Using those diophantine equations, you might be able to predict that some of the numbers aren't going to be prime. Just like you can predict that certain numbers will always be divisible by 3 or by 7; it will save you some computations. But not many I think.

gloom button
Posts: 918
Joined: Fri Sep 28, 2007 5:57 pm
Contact:

Re: Triangles, parallelograms and Shimura curves: Maths!

Yeah, it's really just a sellotaped-up bad idea at this stage. It's back to the drawing board to find something a little prettier, I think.

I've (re)learned a lot about number theory and the distribution of primes though, reading around the problem last night. So that's something.
the trouble with personalities, they're too wrapped up in style
it's too personal; they're in love with their own guile

Who is online

Users browsing this forum: No registered users and 1 guest