Mathematics - 2012-12-23 (page 3 of 3)

Hey all, in the equation x^2 + 8x + 7 = 0, for example, there are 3 x's. The resulting graph would be a parobola, and therefore 2 of the x's fall on the x-intercept. Which 2 x's in the equation fall on the x intercept, and what does the other x represent? It's not as clear as the linear equations where b represented the y-intercept and the m represented the slope. In this quadratic equation, I would like to know what the 3 xs represent.

Khromonkey

what do you mean there are 3 x's?

Charlie

You dont mesn roots do you? Because it only has two

JohnMerlino

x^2 = x*x

or do all the x's represent the same value?

That quadratic equation is the combination of two linear equations

the linear equations make sense to be straight lines

Im just wondering what causes that quadratic to be a parabola and how to understand each of its segments

specifically the x^2 what does that cause on the graph?

Khromonkey

Ok, what do you mean the combination of two linear equations?

JohnMerlino

7:46 PM

for example: x^2 + 5x + 6 = (x + 2)(x + 3)

left is quadratic and right is two linear equations

Khromonkey

ok

JohnMerlino

where 1 is the slope for each of them and 2 and 3 are the y-intercepts

and if you were to graph them, you would get a straight line

and so its easy to understand what each part of the linear equation represents

Khromonkey

yes

JohnMerlino

so now I would like to understand what each part of that quadratic equation represents in reference to a graph

Khromonkey

ok so the general equation of a linear equation is y=mx+b

where m is the slope and b is the y intercept

the general equation for a quadratic equation is y=ax²+bx+c

do you understand this notation?

JohnMerlino

7:49 PM

yes

Khromonkey

Lets start with the simplest: c c is the y intercept

a is the second simplest: a tells you if the parabola opens upwards or downwards and how wide it is.

for example if a is negative the parabola will open down

like a bridge

or an arch

Do you know a little bit of calculus?

JohnMerlino

little bit

pourjour

hi people can someone help me to prove that $$\forall x>0\quad (1-\frac { 1 }{ x } )\sqrt [ 3 ]{ { x }^{ 2 }-2x+2 } \quad \le \quad x-1$$

Charlie

@pourjour !!!

pourjour

@Charlie hi charlie!!

Charlie

7:54 PM

@pourjour how are you?

pourjour

@Charlie fine and u?

JohnMerlino

So in the equation what does the x^2 and bx represent?

Charlie

@pourjour good, good!

pourjour

@Charlie the Christmas!!

Charlie

@pourjour yes yes!!!:)))

pourjour

7:56 PM

:D

$\forall x>0\quad (1-\frac { 1 }{ x } )\sqrt [ 3 ]{ { x }^{ 2 }-2x+2 } \quad \le \quad x-1$

Charlie

@pourjour im on my mobile...

pourjour

@Charlie wow so u're not at home?

Khromonkey

if you differentiate ax²+bx+c you will get 2ax+b. To obtain the vertex you use that the derivative must be 0 in a maximum or minimum. Therefore you set 2ax+b=0 to get x=-b/2a. so the vertex is located in x=-b/2a

Charlie

@pourjour no, not at my place

pourjour

@Charlie so, what are u doing right now?:)

user19161

8:00 PM

@Khromonkey That looks like Jayesh!

Khromonkey

@JasperLoy what does?

Charlie

@pourjour talking to you

user19161

@Khromonkey You must use the left arrow.

user19161

@Charlie Hey M.

pourjour

@Charlie hhhh :D

Charlie

8:01 PM

@skull heeeeeeeyyy!

Khromonkey

@JasperLoy What do you mean i must use the left arrow?

skullpatrol

@Charlie Wazzup?

pourjour

@Charlie when are u going to celebrate Christmas?

Charlie

@JasperLoy hey jas jasper jasper!!!!!!!!!!!

user19161

@Khromonkey Well, you use the right arrow to reply and the left arrow to link backwards. Hover your mouse over the line and you will see...

Charlie

8:02 PM

@skullpatrol everything fine!! You?

user19161

Oh man, they should have these arrows in the chat FAQ.

Charlie

@pourjour tomorrow at night

skullpatrol

@Charlie Fine thanks :)

user19161

Or they should pay me to conduct a how to use chat arrows once a week...

Charlie

@skullpatrol awesome! Long time no see!

Khromonkey

8:03 PM

Ohh you're saying ramanujan looks like jayesh!

user19161

@Khromonkey Did you just wake up?

skullpatrol

@Charlie yep ;)

pourjour

@Charlie nice!!!

@JasperLoy hhhh

Charlie

@Khromonkey but actually doesnt look that much, no...

Khromonkey

No, The problem is I fell asleep too late

pourjour

8:05 PM

@Khromonkey are u talking about the inequality?

Khromonkey

what inequalilty?

user19161

Do people search questions before posting?

user19161

The reference requests are getting silly.

user19161

Every now and then someone comes and asks what is a good book on XXX?

user19161

Then YYY will answer ZZZ.

pourjour

8:06 PM

$\forall x>0\quad (1-\frac { 1 }{ x } )\sqrt [ 3 ]{ { x }^{ 2 }-2x+2 } \quad \le \quad x-1$

user19161

A few days later the same question again...

Khromonkey

That would be me, sorry

pourjour

@robjohn any idea about this (sorry for disturbing u)

Khromonkey

But in all honesty sometimes the questions dont exist on the main page

Allthough if you cant find the question you should ask it yourself

on the main page

I mean as you pointed out it is a very common question @JasperLoy

Charlie

@skullpatrol thats

skullpatrol

8:10 PM

@Charlie Joy for Christmas

Charlie

@skullpatrol that's why i wanted yoy to make that mail...unless of course you don't wanna talk away from the chat...

Khromonkey

Can you help me with an exercise in number theory?

user19161

@Khromonkey Then he is nuts.

Khromonkey

why is p always the largest prime that devides (p-1)!+1?

Charlie

@skullpatrol :) hehehe

pourjour

8:12 PM

any help people?

Khromonkey

the fact p divides (p-1)!+1 is wilsons theorem

Old John

@Khromonkey don't think that is true

user19161

My knowledge of number theory is limited to gcd and lcm, lol.

Khromonkey

sorry: smallest

pourjour

Ok people enjoy Christmas !

Old John

8:14 PM

better!

well - Wilson says that $p$ really is a factor, and any smaller prime gives a remainder of 1

Khromonkey

it is in this book books.google.com.mx/books/about/…

Old John

@Khromonkey Do you follow my 1-line proof?

Khromonkey

yes

Ethan

0

Q: Currious property of monotonic functions

If f(x) is continuous and monontonic on the interval $[1,\infty]$, and $f'(x)\leq\frac{1}{x}$ on the interval $[1,\infty]$, is $$\lim_{n \rightarrow \infty} \frac{1}{nf(n)}\sum_{k=1}^n f(k)=1$$ This is by no means a theorem also, its just a guess I made after expeirmenting with sums of the n...

user19161

@OldJohn HAHAHAHAHA

Khromonkey

8:16 PM

If you know that fact it is obvious

Old John

what fact? Wilson?

Khromonkey

yes

user19161

@Ethan Speaking of which, you haven't accepted my answer on that question. =)

Old John

Oh - don't you know that yet?

Ethan

where did you post the answer

Re-post it their

Old John

8:17 PM

*there

Ethan

there

what ever

user19161

@Ethan Here math.stackexchange.com/questions/261180/…

Ethan

I mean as long as you understand what It is I am saying, no information was lost, in the communication

Khromonkey

I know the following about wilson: p| (p-1)!+1

He proved this using the stellated ngons

@OldJohn

Old John

@Khromonkey that is all you need for my 1-line proof

user19161

8:19 PM

Is there an Old John's theorem on primes?

Khromonkey

Im an idiot

i got it

what is wrong with me?

Old John

@JasperLoy No - only an old john theorem on asymptotic values of fine continuous functions :)

Charlie

@Khromonkey nothing is wrong...it happens

Khromonkey

thanks anyways

user19161

@OldJohn Ah, that potential theory thing!

Old John

8:21 PM

@JasperLoy yep :)

Khromonkey

My brain is more valuable by its calories than by its intellectual worth

user19161

I feel like answering all the reference request questions here, since I know the titles of over 9000 books!

Old John

@Khromonkey I think that you could equally deduce Wilson's theorem from your result about the smallest prime factor - so proving your result without using Wilson is probably as hard as proving Wilson's theorem

user19161

But the reason why I don't answer them is because I don't know which book to recommend the asker, since there are over 9000!

Khromonkey

I have 45 math books

Old John

8:23 PM

@JasperLoy The other day, for the first time ever, I quoted something in an answer from page 1 of a book ... that must be a rarity, as books almost always spend the first few pages reminding readers of stuff they ought to know

Khromonkey

I am like the math library for my frined

friends

have you read diestels book in graph theory?

user19161

@OldJohn Page 1 usually has many words and few symbols.

Khromonkey

unless its diestels graph theory

user19161

@Khromonkey That's a standard text.

Khromonkey

I read a little of it in the library

the first page is like super condensed

Old John

8:25 PM

@JasperLoy yep - or just a list of symbols they are going to use - my quote is here

user19161

@Khromonkey Do you know what is the most condensed book I have come across???

user19161

It is...

Khromonkey

not yet

user19161

Federer's Geometric Measure Theory!

user19161

HAHAHAHAHAHAHA

user19161

8:27 PM

When you see that book you will die!

Khromonkey

how am i supposed to read that laugh?

user19161

It is so condensed you will burst out laughing when you see it.

Old John

@JasperLoy "Basic Number Theory" is even worse than Federer

Khromonkey

At first glance I read it like it was super evil and accompanied by bachs toccata in D

user19161

I intend to write like Federer in future.

user19161

8:29 PM

@OldJohn Yes, a very misleading title!

Old John

@JasperLoy Serre's "A Course in Arithmetic" is also a bit misleading - but is more in the way of a joke, I think

Khromonkey

amazon.com/Basic-Number-Theory-Classics-Mathematics/…

user19161

@Khromonkey Check out Federer too.

Khromonkey

amazon.com/Roger-Federer-Greatest-Chris-Bowers/dp/1843583895/…

@JasperLoy I didnt think you where into that

user19161

@Khromonkey Hahahaha, you know nothing about me yet!

Charlie

8:33 PM

@Khromonkey i was thinking about that too ...

user19161

My favourite tennis star is...

user19161

Nadal!

Charlie

@JasperLoy hey jas have you seen Aaron???

user19161

@Charlie He went to Florida.

Khromonkey

@ Nadal is great but he is very injured

user19161

8:36 PM

@Khromonkey Like me, my soul is injured...

Khromonkey

Why?

user19161

That is my secret...

MSEoris

guys i cant stay too long but i have to share this wonderful song with you

►

Charlie

@JasperLoy I KNOW!!!!he told me that somedays ago...but he appeared here once and he doesn't answers my mails... maybe he doesn't have internet now...

user19161

@Charlie Yeah, no hurry. He's probably chatting up some girl on the beaches of Florida. =)

user19161

8:39 PM

@MSEoris I think I have seen this before.

Khromonkey

@MSEoris Sublime, marvelous, simply breathtaking!

MSEoris

indeed, its perfect nerd humor :) i love it

user19161

I'm sorry, I think nerd humour is not funny!!!

user19161

That's why it's funny!!!

Charlie

@JasperLoy yeah maybe. he's a nice boy :)

MSEoris

8:41 PM

there are certainly many layers of interpretations, but even without the humor the puns alone make it wonderful

user19161

The thing I find weird about some questions here is: their proofs can be found in many textbooks, why ask here???

Khromonkey

I live in a developing country @JasperLoy

Old John

@JasperLoy laziness?

Khromonkey

Instead of libraries we have telenovelas

user19161

@Khromonkey I thought you said you were from the US?

Khromonkey

8:44 PM

Im from mexico

user19161

@Khromonkey Ah OK.

user19161

Anyway, some countries can have very tall buildings but be very primitive otherwise...

Khromonkey

I said I wanted to go to college in usa

We have tall buildings

user19161

So, don't judge a country by its buildings...

Khromonkey

we just dont have libraries

Jayesh Badwaik

8:45 PM

@JasperLoy textbooks are expensive (too) many times.

user19161

Some universities may look very beautiful, but have shit lecturers and shit syllabuses.

Jayesh Badwaik

@Khromonkey true that. infact apart from my university library, I have never gone to any other library ever.

@Charlie Hi!

Charlie

@Khromonkey mexican telenovelas are famous here too

user19161

@JayeshBadwaik Therefore, library.

Charlie

@JayeshBadwaik hi!!!

Khromonkey

8:47 PM

There is a pretty big library in my school, it has books on measure theory, algebraic number theory and it even has some journals on super advanced stuff

user19161

Gee, this Marvis guy types so fast...

Khromonkey

However the people who run it are nuts

user19161

@Khromonkey Are you in a university now?

Khromonkey

There is not a single book on topology

Charlie

@JayeshBadwaik :*

Khromonkey

8:48 PM

No, I go to en.wikipedia.org/wiki/ITESM

user19161

@Khromonkey If you are in high school, the library there should not have these books.

Khromonkey

It also has high school

user19161

Probably the librarian doesn't know the level of the books and thought they were elementary!

Khromonkey

I go to a building with high school and university

user19161

OIC.

user19161

8:49 PM

Sometimes people mistake "Algebra" for x+x=2x.

4

Khromonkey

They have books on manifold theory but not a single rigurous book in calculus

Jayesh Badwaik

@JasperLoy yes, but dude, my university is one of the 25 good libraries in the whole of india? with the access restricted to around 100,000 students and some university staff every year? number of libraries are too less.

Khromonkey

unless you consider stewart or calculus for dummies rigorous

Charlie

@jayesh

Jayesh Badwaik

@Charlie wassup?

user19161

8:50 PM

@JayeshBadwaik too few, not too less.

user19161

@Khromonkey No way!

Khromonkey

If I want to go to a public library with math books I have to drive one hour.

user19161

@khro Just ask me for book recommendations of any topic!

Charlie

@JayeshBadwaik I'm good! You? You've been talking to someone...

:P

Khromonkey

Besides, my math teachers are all like businesspeople

Jayesh Badwaik

8:52 PM

@Charlie yes, I was nervous. :P

user19161

Most of the math teachers I have come across don't really understand the math they teach.

Old John

@JasperLoy I am getting into an edit war on this - someone keeps trying to delete the first "converges!

Khromonkey

Half of the time is spent on: we are great investors we take inniciative

Charlie

@JayeshBadwaik oh! Why?

Haha

Khromonkey

my friend had to explain to the teacher that a graph of a vertical line was not a function from x to y

user19161

8:54 PM

@OldJohn I am thinking of other possibilities. If two people edit at the same time the greater edit will override the lesser one, but this does not seem the case here.

Khromonkey

He had to teach him the vertical line test

Old John

@JasperLoy Yeah - I think I am right in saying that the sentence needs the word converges twice :)

user19161

@OldJohn Yes, this is grade 1 English!

user19161

@Khromonkey Most of education is a waste of time.

Khromonkey

I have your back @OldJohn

user19161

8:57 PM

They put 16 year olds in schools and teach then how to solve quadratic equations when they should be having fun instead, lol.

Old John

@Khromonkey Thank you!

Khromonkey

no problem

Mike Spivey

Is what I'm asking in this question sufficiently clear? In particular, is it clear why Robert Israel's answer isn't really what I'm looking for?

Khromonkey

I have to go guys

user19161

They try to justify why quadratic equations are impt, not knowing that most of these kids will never use one in their life.

Khromonkey

8:58 PM

yes

Well see you later guys

@OldJohn thanks for your help in the wilson problem

Old John

@Khromonkey No problem!

Just noticed that condensation problem is a duplicate - I will vote to close ...

Hi @WillJagy

Jayesh Badwaik

@JasperLoy what if a 16 year old thinks solving quadratic is fun?

user19161

@JayeshBadwaik Ah I know someone would start a whole debate on this. I am tired, I need to sleep. Over and out!

Old John

@JasperLoy g'night!

Mike Spivey

Random question, for anyone who feels like answering it: How did you first find out about math stack exchange?

Old John

9:06 PM

@MikeSpivey I first found MSE by visiting MO and seeing people being sent in the direction of MSE

Mike Spivey

@OldJohn That was the same for me. Did/do you spend much time on MO?

Old John

@MikeSpivey Not plucked up courage to ask any questions there yet :)

But I keep an eye on it

Mike Spivey

@OldJohn I understand the feeling. :)

Old John

@MikeSpivey :)

Jayesh Badwaik

@MikeSpivey I found stackoverflow first, and about six months later, did I get to MSE.

Mike Spivey

9:10 PM

@JayeshBadwaik Are you a programmer?

Jayesh Badwaik

@MikeSpivey No, electronics engineer. But I was programming then (simulation code), and a lot of really good answers on google search seemed to be present on stackoverflow.

Mike Spivey

@JayeshBadwaik I spent the spring working for an investment firm, where I did a lot of coding in Matlab. I found a lot of good answers to my Matlab questions on stack overflow as well.

Jayesh Badwaik

interesting. for me it was c++.

Will Jagy

9:44 PM

@OldJohn Alright. Why did Mourinho really bench Casillas yesterday? Real Madrid are now completely and utterly fucked.

Old John

@WillJagy I don't know, I'm afraid - not enough time in my day to follow all the football I would like, unfortunately

@WillJagy Did the attachment finally arrive?

pourjour

@JayeshBadwaik what are u most great project in c++?

Will Jagy

@OldJohn, yes, the other page from Pommeranian. Really good book. I looked up the octonions, that strange integer matrix problem works for any vector length in dimension 2,4,8, and Jones and Pall manged to drop from dimension 4 to 3 when the length is an integer. So, maybe, 8 can drop to 7,6,5, but may be a shitload of work.

Old John

@WillJagy but by "Jagy's theorem" we know it fails for $n^2$ for $n=3, 5, \dots$ :)

Will Jagy

10:01 PM

@OldJohn, exactly so. There are some famous problems in integer quadratic forms which are true up to some dimension and then false. The best known is that, up to dimension 7, if a positive symmetric and integral matrix has determinant 1, it is integrally equivalent to the identity, that is can be written as P P' where P is integral and P' its transpose. But in dimension 8 we suddenly get E_8, see Gram matrix at math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html

Oi @ian !!

@Charlie Oi!

@IanMateus e aí?

@Charlie tô bem, como vai ser o teu natal?

Charlie

@IanMateus como sempre :P

Ian Mateus

10:07 PM

@Charlie como é? heh

Old John

Good grief! there is a "Catalogue of Lattices"!!
Always been fascinated by how things change as we increase the dimension ... [like this one](http://math.stackexchange.com/a/258558/32441) - interesting (but purely coincidental, I guess) that things with spheres change at 9 dimensions too ...

Will Jagy

Yes, Gabriele Nebe is a big name in quadratic forms. The main tradition is in Germany, I think it is fair to say that.

Old John

@WillJagy Carrying the torch lit by gauss and some others around that time :)

160,000 lattices in their database - that is insane ...

Charlie

@IanMateus a gente se reúne, come :P troca presentes, essas coisas

Old John

You were spot on with your suggestion of lacunary series in that old answer :)

Will Jagy

10:19 PM

Lacunary series? I would never say that in public, I was brought up properly.

Old John

@WillJagy I apologise profusely ... and promise never to use such a foul name for a variable in a C++ program ... :)

Will Jagy

@OldJohn, I'm glad you have seen reason. I think words such as smegma, spooge, and so on, have no place in polite discourse. What old answer did you mean? Or, hmmm, there may be a joke i have entirely failed to understand. Or maybe a reference to Charlie and Ian Mateus, who are conversing in some invented language

Charlie

@WillJagy it's PORTUGUESE

Will Jagy

@Charlie, well, you may be telling the truth. In that case, you tell me why Jose Mourinho benched Iker Cassilas yesterday against Malaga.

Charlie

10:35 PM

@WillJagy I don't like soccer

Old John

@WillJagy It was in a comment to this question where you used the dreaded word ... I win!

Justin

Hello

Charlie

hello

@Justin wassup?

Will Jagy

@OldJohn A touch, I do confess it! I fear I breathe my last.

Old John

@WillJagy I promise not to reveal your lapse to anyone ... other than the chatters present here ...

... and the readers of MSE who actually read comments ...

@WillJagy so ... you are a number theorist who delves into analysis sometimes, and I am an analyst (of sorts) who likes to delve into number theory - are we anti-matter versions of each other???

Will Jagy

10:45 PM

@OldJohn, I'm not supposed to speak any more, being officially dead. I see why you expected I would recall that comment, it is part of the Pommeranian question we had been discussing. Let's see, the complex analysis bit was a sequence of questions on MO about solving f(f(x)) = g(x) for a fixed function g(x), near a fixed point g(a) = a. The short version is that f(x) is almost never holomorphic, or defineable, around a.

Old John

@WillJagy That sounds interesting and actually no longer surprises me - a bit like the fact that almost all continuous curves in $\mathbb{C}$ are actually non-smooth - like brownian motion

well - ok - it does surprise me a bit ...

Will Jagy

@OldJohn, for the quick version see mathoverflow.net/questions/45608/… with a fair percentage of published (and responsible) background at zakuski.utsa.edu/~jagy/other.html It is an interesting topic that has attracted any number of irresponsible blog posts

Old John

@WillJagy It looks extremely interesting - but I think I need to stop myself form thinking I should get too involved in it, having struggled to read Carleson and Gamelin's book on complex dynamics. There are too many fascinating areas of maths to look into - can I ask for another whole lifetime to learn?

Charlie

11:00 PM

@OldJohn you said everything john...

Old John

@Charlie I did???

@Charlie ah - "another whole lifetime to learn"? a quote from a Joan Baez song, I think :)

Will Jagy

@OldJohn, you may ask. Anyway, it is a pretty trivial and very small part of dynamics. The whole story is a trick due to Jean Ecalle in his dissertation, approximately 1970, which he was rude enough to write in French. The way he does it almost always gives a term involving log(z-a) so is defined on a sector.

Old John

@WillJagy Do you have a handy reference for deriving a parametrised solution of $a^3=b^2+c^2$ using gaussian integers? - I am sure it must be elementary, but had too much whisky tonight to do it :)

wj32

hi

i've been doing some tedious exercises

what's this kind of thing called?

Old John

11:16 PM

Abstract nonsense? (aka category theory)

Will Jagy

@OldJohn, I am not so sure a parametri(sz)ed solution is easy. I can tell you that there are solutions for a given a if and only if, for any (positive) prime q == 3 mod 4, then v_q(a) is even, meaning the exponent of q in the prime factori(sz)ation of a is even. If so, all solutions can be produced using the factori(sz)ation. Parameters, maybe, maybe not.

wj32

@OldJohn: is there some way to make it less tedious?

it seems like i'm just shuffling symbols around

and searching for domains/codomains that "match up"

Old John

@wj32 There is a cure ... take up analysis :)

wj32

i have even less motivation when i'm studying analysis :(

Old John

@wj32 I gave up on algebra after spending most of a day chasing elements round diagrams in a five-lemma situation, and realising I was never going to win

wj32

11:19 PM

ok what i meant was, is there some way to make sense of what i'm doing

Old John

It was too much like wack-a-mole

wj32

right now it seems like black magic

it always seems to magically work out in the end

Old John

@wj32 Sorry - I am being facetious - I should let someone who knows answer. I guess the best I could say is "sleep on it" it will probably look better tomorrow ...

wj32

ok thanks, i might try asking on the main site

Old John

@WillJagy I think this works: $a=m^3-3m n^2$, $b=3m^2n-n^3$, $c=m^2+n^2$, and I am sure it can be obtained by looking at factorisations in Gaussian integers - but getting confused

@wj32 Probably a good idea - if you can phrase it into a good question, it should get some attention from people who are not as drunk as me :)

Will Jagy

11:26 PM

@OldJohn, give me a minute. Just remember that your "a" must be the sum of two squares.

@OldJohn, you were close. If you change letters to now write a^2 + b^2 = c^3, one set of solutions is a = u^3 - 3 u v^2, b = u^2 v + v^3; c = u^2 + v^2. These are by no means all solutions, although it does identify all possible "c".

Old John

@WillJagy should your expression for $b$ have a minus sign instead of a plus?

Will Jagy

@OldJohn, sorry. One method gives a = u^3 + u v^2, b = u^2 v + v^3, c = u^2 + v^2. The other one is yours, a = u^3 - 3 u v^2, b = 3 u^2 v - v^3, c = u^2 + v^2.

Old John

11:41 PM

@WillJagy Thanks, Will - I will look at it again tomorrow, when I hope to have a clearer paleolithic mind :)

@WillJagy Oh - have you seen how to get LaTeX working in this chatroom?

Will Jagy

@OldJohn, yes, those give two families. However, the number of different representations for a given c = u^2 + v^2, and for that matter such c^3, increases depending on the number of distinct prime factors p == 1 mod 4. An explicit value is known. the exponents on the other prime factors do not increase the number of expressions.

Old John

@WillJagy Tricky little suckers, these diophantine equations!

Will Jagy

@OldJohn, no, but I see on the right column there is a post by anon about Latex support for chat, on 18 December

@OldJohn, do you want all solutions to a^2 + b^2 = c^3? I have my doubts about a single formula.

Old John

@WillJagy There is some strange process (I did it once) - you drag something to the "bookmark bar" in Chrome or Firefox and when you click on it, then all the LaTeX in the chat becomes readable - you only have to click the "thing" once per session - there are some instructions in anon's link, I think

Will Jagy

@OldJohn, if I've got to I will make a Latex pdf and email you that. As far as chat, I want to keep my Latex virginity.

Old John

11:50 PM

@WillJagy No - I am not desperate for a formula for all solutions - what is really bothering me is that I can't work out how to derive my particular parametrisation using Gaussian integers - it must be something easy, since all the bits look like parts of the expansion of something like $(x+iy)^3$

I think I should resign myself to the fact that I am going to achieve nothing useful this side of Christmas :)

Will Jagy

@OldJohn, if N(z) = z bar(z) is the squared magnitude, then N(z^3) = (N(z))^3. Take z = x + y i

Old John

@WillJagy Yep - I got that far OK ...

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$Mathematics$ Mathematics