Makes me wonder, for what polynomials $p_i$ does $x_1,\cdots,x_n$ being pairwise coprime entail that $(p_1(x_1,\cdots,x_n),p_2(x_1,\cdots,x_n),\cdots)$ only takes on finitely many values

@robjohn Right. Bezout all over the place.

So, the problem is

$$\left( {a,b} \right) = \left( {c,d} \right) = 1 \wedge \frac{a}{b} + \frac{c}{d} = n \in {\Bbb Z} \Rightarrow \left| d \right| = \left| b \right|$$

@Eugene Huh, very useful!

Don't tell me anything

I'll write the proof.

so i can find the library in spanish speaking countries and a writing table in germany.

i'm well equipped to travel around the world!

@Eugene That's great.

hey guys whats up! received my swag today. Unfortunately very less number of stickers :(

@Eugene I'll accompany you in case you want to find the writing table in spanish speaking countries and the library in Germany.

@Gigili that just made my day awesome

@PeterTamaroff huh i guess i did prove it yesterday!

@robjohn The gcd is special case $f(x) = x^3$ of $\ \left(a-b,\dfrac{f(a)-f(b)}{a-b}\right) = (a-b,f'(a))\ $ e.g. see here. Good luck Bezouting that!

@Peter @anon @Eugene see above

@BillDubuque Sheesh, are those $\gcd$s?

That's very nice. Looks Hensellian.

well @robjohn? are you going to take @BillDubuque's jab lying down?

@PeterTamaroff Yes, you can differentiate in number theory, as long as you do it universally!

@robjohn: and hi to you!

@BillDubuque Interesting! We'll maybe talk in equal terms some day. BTW, is my last proof OK?

I'm so hungry! I didn't have a lunch today. Let me cook a delicious dinner

Reminds me of this question, .. ish.

@Eugene I will have to lie down for a while. My wife is waiting to go to lunch. I will look into it when I get back :-)

BBL!

@robjohn bye!

@anon Did the multiple ping above notify you, or is chat limited to 1 ping like main?

@Ilya lets you cook a delicious dinner What's it?

No, ping in chat is not limited by number of people.

it notifies me as well

It is however limited by number of characters in one's handle (so JM and tb never get pinged).

This is infuriating, is there some rule for how to write down an answer for a nintegral involving trig subsitution? There are an infinite numbers of ways to write the same things involving trig identities, how do I know which is the correct way?

i'm trying to put off grading as long as possible. until next semester hopefully.

@Jordan "Hello" is usually welcome.

I didn't think anyone really cared for me here

@Jordan I already told you: there is no right way.

You need to get experience

So as to realize which one will work faster or easier.

haha

exactly

But there isn't a way.

just now we found 4 ways to prove @PeterTamaroff's question

Well I just meant to get the answer from my book, wolfram and my book had the same form of the answer and I did not realize it was an equivilent statement to waht I had so I did the problem over four times before I figured out I can manipulate what I had to equal what they have

@Eugene Wait, the one of number theory or the one on $$\frac{s_n(a+1)}{ns_n(a)}$$?

number theory

i didn't see that one

Jordan try not to get frustrated. every message you leave is about your own perceived inabilities to solve the question. If you are not understanding you have to keep heading back until you get to the root of the problem and try to solve it. It's important to understand that you can do it, because mathematics is a logical process and you use logic in your daily life

so you will succeed, so believe in yourself

I figured people cannot ask good questions at weekends.

@Gigili that's an idea.

@Gigili i can muster one up if you want?

yeah maybe there are not so many answerers around. I answered a question for the first time and I even got an upvote haha. feels warm and fuzzy

this is why i sometimes prefer MSE over MO. it's more welcoming at times.

@Eugene At almost all times!

I am scared of MO

@BillDubuque well i wouldn't say that exactly.

@Eugene Depends on the one.

@Gigili pardon?

Hum?

This is worth remembering.

@Gigili i didn't understand what you meant by depends on the one

@MattN Taste? I never use hookrightarrow. Rightarrowtail is more symmetric with twoheadrightarrow.

@BillDubuque i've certainly gotten some non-brusque replies on MO.

@Eugene Me neither.

Hello.

@Gigili you are rather funny.

@Eugene Sure, I was exaggerating a bit. But I think it says a lot that we don't get accused here of being "snobbish" anywhere near as often as does MO. Some of us worked hard in the early days to ensure the site was as welcoming as possible to folks of all mathematical persuasions.

@BillDubuque i agree. i think it paid off. thanks for your hard work. i certainly enjoy coming here for the welcoming atmosphere

hello

@tb Nooooo! I like hookrightarrow. </3

has anyone seen this before?

@BillDubuque Do you think Apostol's Intro to ANT is a good book or is it a little outdated?

We were discussing this with @Eugene

His professor uses it for the ANT course, but changed the order of the chapters.

@Eugene I'm quite pleased that most of my x's are indeterminate!

@BillDubuque Here's how he gives it.

@MattN: !

@Ilya What? Is this a cigarette?

@MattN: gtalk

Hello, @Henning.

I don't like when I'm pinged and then got ignored

@Eugene I always liked this one

@HenningMakholm: hi

Hi, Gigili and Ilya.

@PeterTamaroff that is pretty funny!

@Ilya Or perhaps a toothpick because it looks cool? : ) (sorry, if I ignored you I didn't do it on purpose)

@Bill: how are you? And I didn't say hi, sorry. Hi!

@PeterTamaroff you'll be well off with apostol until you learn more!

@Eugene I suspect so, but I guess @Bill can give me good advice.

@Ilya Hi, I'm good, it's always fun to differentiate numbers...

@BillDubuque do you mean the problem you've just discussed?

does anyone here know why we would come up with $3$ descent for computing rank when we already have $2$ descent?

@Ilya Yes.

@BillDubuque icic

@BillDubuque "Do you think Apostol's Intro to ANT is a good book or is it a little outdated? We were discussing this with Eugene - his professor uses it for the ANT course, but changed the order of the chapters."

Is the abstract-algebra tag here really appropriate? I think it's not.

@PeterTamaroff I have not looked at Apostol in ages. I don't have much of a taste for analytic number theory (vs. algebraic).

"not" perhaps being a slight exaggeration.

@BillDubuque Oh. What does Algebraic Number Theory focus on?

@PeterTamaroff class numbers?

I thought the prevailing field was the Analytic Number Therory, pardon my ignorance!

@Eugene Class?

As in "is not in this set"?

@PeterTamaroff it is definitely not

Algebraic number theory sometimes sounds more like geometric number theory, really. :p

Number theory is largely the study of number fields and things associated them.

@PeterTamaroff class numbers

@ZhenLin while certainly true in most respects, it think it's less geometric than say arithmetic geometry

@PeterTamaroff Browse some textbooks titled Algebraic Number theory. Much of the early work was inspired by finding higher degree analogs of quadratic reciprocity (higher reciprocity laws). In fact this motivated much of the early development of abstract algebra.

@Eugene Bah, far from my scope. For now...

My essay supervisor defined number theory as anything that helps us understand $\textrm{Gal}(\bar{\mathbb{Q}} \mid \mathbb{Q})$.

@BillDubuque: I still don't know what quadratic reciprocity is. Hah!

@PeterTamaroff in fact a lot of algebraic number theory machinery was developed to solve fermat's last theorem

What is $\textrm{Gal}(\bar{\mathbb{Q}} \mid \mathbb{Q})$?

@ZhenLin See the very interesting interview with Richard Taylor here.

I've seen claims that an overwhelming fraction of modern algebra wouldn't even exist were it not for attempts to prove Fermat's last theorem.

@BillDubuque What a coincidence, he's my supervisor's supervisor!

when i think about how simple a statement is there cannot be coprime $x,y,z$ such that $xyz \neq 0$ satisfying $x^n + y^n = z^n$ for $n \geq 3$ i'm surprised by the sheer amount of math it's generated

anyway

I think "every surjection has a right inverse" is also very simple, yet it cannot be proven... :p

any bets for 2014 fields medalists?

my money is on bhargava

Porton.

quadratic reciprocity is nice

@HenningMakholm he's gunning for the abel prize

Porton for sure

well his work does solve 17 open problems

@Eugene Yes, but it would be smarter to go for Fields first; there's an age limit on that one.

@Eugene Porton's? Who is he?

porton

@HenningMakholm That's romanticized history - far from the truth All of the major players were motivated much more by higher reciprocity than by an isolated result like FLT, e.g. see the links here for discussion by Franz Lemmermeyer and I (at least for algebraic number theory)

@PeterTamaroff He's also here

@HenningMakholm i guess you're right. can't beat his funcoids and reloids

@HenningMakholm And here.

poor guy

Why is his work so important?

@PeterTamaroff because it solves 17 open problems

@Eugene Which ones?

beats me he created them all