Mathematics - 2012-06-13 (page 3 of 3)

Eugene

21:00

@Gigili i figured.

Gigili

3 vs. 1? WHY?

anon

William's was 12 seconds earlier?

Gigili

I'll something you all. I can't tell because I'll get suspended.

Peter Tamaroff

@anon Hm. But it seems pairwise coprimality is not needed. Just that $(x_1,x_2,\dots, x_n)=1$ and that $x_i$ doesn't repeat, or repeats $k \le |x_i|$ times.

robjohn

@Gigili william answered a few seconds earlier...

Gigili

21:02

@anon 12 seconds? rly?

Someone posted an answer 3 minutes earlier than me but edited it over a million times.

anon

@PeterTamaroff My version requires that $x_n$ is coprime with each of $x_1,x_2,\cdots,x_{n-1}$. Of course for the induction hypothesis to hold we want $x_{n-1}$ coprime with $x_{n-2},\cdots$ and so forth, which amounts to the full requirement of pairwise coprimality.

Peter Tamaroff

Errata in there $\le$ is supp to be $<$.

@anon Ugh, right.

anon

I'm going to write this as an answer.

Peter Tamaroff

@anon Wait, please.

anon

I WAIT FOR NO ONE MORTAL.

Eugene

21:05

another one down! that makes 2 unanswered EC questions i disposed off.

Peter Tamaroff

@anon Fool. I'm no mortal.

@anon Please, do wait, listen (well, read).

anon

k

2

robjohn

@Gigili It's now 3 vs 2 ;-)

Peter Tamaroff

@anon I need to establish a condition on the multiset we were talking.

For example,

anon

$0\not\in S$

Peter Tamaroff

21:06

@anon Sure. But

$$1 + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{3}$$ is clearly not an integer.

anon

but the hard mode version is not your question

Eugene

@robjohn i've decided to dispose of old elliptic curve questions rather than race against arturo and brian.

Peter Tamaroff

Or

Eugene

those are lost causes.

Peter Tamaroff

$$1 + \frac{1}{4} + \frac{1}{2} + \frac{1}{9} + \frac{1}{3} + \frac{1}{{14}}$$ isn't either.

robjohn

21:07

@Eugene probably a wise move, but sad.

Peter Tamaroff

However, $9$ and $3$ are not coprime.

Same with $4$, $4$, $4$.

anon

What's your point?

Eugene

@robjohn well i don't think i can compete with them unless i raise my typing speed by a factor of 50

Peter Tamaroff

@anon That coprimality is not needed.

anon

Yes, I know. What of it?

robjohn

21:08

@Eugene they have to sleep sometime! :-)

Peter Tamaroff

@anon That then induction falls apart... ='(

Eugene

@robjohn not from what i've seen!

anon

Indeed.

Peter Tamaroff

But have the idea, I just can't make it into a proof.

Darn.

robjohn

@PeterTamaroff practice and experience helps there.

Peter Tamaroff

21:09

@anon Even if we summed $4$ $1/4$ths, summing a $1/9$th would make the sum a non integer.

Eugene

@PeterTamaroff indeed.

Peter Tamaroff

@Eugene I need to think about the relations between the numbers in the set.

robjohn

@Eugene shhh.. they'll start to suspect

Peter Tamaroff

I think the order is important.

Eugene

@robjohn sorry!

Peter Tamaroff

21:10

Also the number of repetitions, and coprimality.

So many variables!

Eugene

welcome to number theory! it only gets worse.

Peter Tamaroff

@Eugene Yay!

Challenge accepted, dude.

robjohn

The characterization of an integer being the sum of two squares is a good start in number theory, I think.

I think it got me my first bounty here

Peter Tamaroff

@robjohn Don't you mean $4$ squares?

robjohn

@PeterTamaroff is there anything that can't be written as 4 squares?

Peter Tamaroff

21:13

@robjohn Dunno. But $3 \neq a^2+b^2$.

Eugene

@PeterTamaroff no

robjohn

@PeterTamaroff yes, but proving the characterization of when an integer is the sum of two squares is interesting and a good introductory topic

Peter Tamaroff

@robjohn Oh! OK. Couldn't get your point.

anon

quadratic reciprocity is a good gateway into number theory

Eugene

@DylanMoreland i worked out the elliptic curve problems!

robjohn

21:15

for example, $x^2+y^2\not=76$

Peter Tamaroff

@Eugene I'll keep thinking about the theorem. Maybe later on I'll prove it.

Zhen Lin

I find Diophantine equations more interesting than quadratic reciprocity...

Eugene

@anon reciprocity in general motivates so many questions in NT

Peter Tamaroff

@Eugene BTW, I reserved this book in a bookshop today amazon.com/God-Created-Integers-Mathematical-Breakthroughs/dp/…

anon

I don't like Diophantine equations :(

Peter Tamaroff

21:16

Amazing content.

Eugene

@ZhenLin diophantine equations are a generalization of reciprocity.

skullpatrol

Do the users who answer questions get updates if the OP edits the question?

Eugene

@anon i love em!

Peter Tamaroff

@skullpatrol Nuuu.

Zhen Lin

@Eugene So, can you tell me why reciprocity is interesting?

Gigili

21:18

I don't get it. You can say who answers first before an hour but not after it?

skullpatrol

@PeterTamaroff Then the OP could almost completely change the question from the original.

Peter Tamaroff

@skullpatrol Yes. But he/she should tell about an edit.

Eugene

@ZhenLin you can see this

Américo Tavares

@robjohn @robjohn Thanks for your explanation and for your answer to my question on the main site.

robjohn

@AméricoTavares any time. Thank you for the points :-)

Eugene

21:21

@ZhenLin a lot of algebraic number theory machinery was motivated by reciprocity

Zhen Lin

@Eugene Anyone can point to a fact. Why is this fact more interesting than any other? :p

robjohn

@AméricoTavares Find a proof of the Euler-Maclaurin Sum Series and you'll see the integral that gives the remainder term.

Américo Tavares

@robjohn Your answer was easier for me than any other.

skullpatrol

@robjohn I almost got suspended for adding in the "textbook answer" to my question :(

Eugene

@ZhenLin can you be more specific? it almost seems like you're asking why some math is more interesting than any other...

anon

21:22

@PeterTamaroff Why are you changing your question so drastically?

Zhen Lin

I'm basically a layperson when it comes to number theory. I've never taken a basic course.

robjohn

@skullpatrol how did they justify that? Were they worried about plagiarism?

anon

Your theorem applies to pairwise coprime integers just fine.

Eugene

@ZhenLin reciprocity is a useful piece of machinery actually

Zhen Lin

OK, can you give an example?

Eugene

21:24

@ZhenLin for instance you can count the number of points on certain elliptic curves using legendre's symbol.

skullpatrol

@robjohn I'm not sure, I wanted answers from those users who showed interest in the question, so I put it into the "comments" section of each of their answers. The mods said this was SPAM.

robjohn

@skullpatrol where?

Eugene

@ZhenLin in chapter 8 of ireland and rosen, they show how you can count the number of solutions of diophantine equations over finite fields using gauss and jacobi symbols!

Gigili

@skullpatrol You wanted them to change their answers? That's not how it works.

Zhen Lin

@Eugene OK, now I have some idea of what reciprocity is useful for. Thanks!

robjohn

21:27

@skullpatrol You certainly got a lot of attention to the question.

Eugene

@ZhenLin it's tedious though if you've only seen quadratic residues in an elementary number theory course though. then it looks like it's developed for the sake of being developed.

@ZhenLin also you can use it to prove interesting stuff about taxicab numbers

Zhen Lin

I mean, before, the only thing I knew about counting points on an elliptic curve was to use the Frobenius action over finite fields and invoke the Lefschetz trace formula :p

skullpatrol

@robjohn I'm thinking of putting a bounty on it :D

robjohn

@skullpatrol negative 50 points or minus 50 points?

skullpatrol

@robjohn Both :D

robjohn

21:29

answer this question and risk losing 50 points

skullpatrol

Is that how bounties work?

robjohn

@skullpatrol no, you can't have a negative bounty

Américo Tavares

@robjohn I'll see it in detail. I've read long time ago the evaluation of $\zeta (s)$ by the Euler-Maclaurin Sum in Edwards' Riemann's Zeta Function (p. 114), and one or two more computations, but it is actually something I still have to understand better and be able to apply by myself.

robjohn

@AméricoTavares Good book! I have it. I need to reread some of it I think.

Peter Tamaroff

@anon I just added that the numbers are coprime.

skullpatrol

21:32

@robjohn What amount of a bounty do you suggest, oh great mean square bounty hunter?

robjohn

@AméricoTavares I find the EMS formula quite handy, indeed!

Peter Tamaroff

@anon Happy to hear that! But maybe it can be applied to the harmonic numbers

We just need to split the harmonic number approprietly and apply the theorem.

anon

No, that won't work.

robjohn

@skullpatrol I got a bounty this morning that I had not noticed before, thanks to Américo. I don't know if you can put a bounty on a CW question.

Eugene

@ZhenLin you can see here to see how ken ono uses the legendre symbol to count the number of points on certain classes of elliptic curves with CM over finite fields

skullpatrol

21:34

@robjohn But I'm not satisfied with any of the answers :(

Peter Tamaroff

$$\eqalign{
& 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} = \cr
& 1 + \left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{6}} \right) + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} = \cr
& 1 + \frac{1}{2}\left( {1 + \frac{1}{2} + \frac{1}{3}} \right) + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} = \cr} $$

Eugene

@ZhenLin also Jacobi symbols are a very canonical example of dirichlet characters!

anon

@PeterTamaroff You can split the harmonic number into a sum of sums, wherein the theorem applies to each individual sum, hence write the harmonic number as a sum of nonintegers, but that doesn't establish that the harmonic sum is itself noninteger.

Zhen Lin

aaaaa, lots of words I don't know!

Américo Tavares

@robjohn It is proved in Apostol's Mathematical Analysis.

robjohn

21:35

@skullpatrol You can try to put a bounty and see if it works. As for how much, that is up to you.

Peter Tamaroff

The number not in parenthesis is a sum of exclusively coprime numbers.

Eugene

@ZhenLin it's a rich topic actually. BUT... i like diophantine equations better

robjohn

@AméricoTavares It is proven in a number of places. It is similar to the proof of Taylor's theorem (power series)

@AméricoTavares repeated integration by parts.

Peter Tamaroff

@anon I'll try to make an extension then.

Eugene

@ZhenLin so many more beautiful results in DE!

and i don't mean differential equations

robjohn

21:38

I have to leave my nice sunny courtyard here and walk to the library where the exam is being held that I am proctoring. See you all later!

skullpatrol

later

Eugene

@robjohn you too eh? good luck then! bye!

Américo Tavares

@robjohn Just a final question. Since a duplication of my question is still not answered on MathOverflow, do you think I can link to your answer there?

@robjohn See you later!

Jonas Teuwen

22:41

@robjohn Hi!

Jordan

23:09

oh lawd I is not gettin differential equations

anon

23:27

400 points?? skullpatrol, you are totally screwy

Jonas Teuwen

This place is infested with harmonic analysts. Why the fokke are they working at 1:40AM?

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$Mathematics$ Mathematics