Mathematics - 2017-05-20 (page 1 of 5)

Dodsy

00:00

Genius Balarka.

Balarka Sen

(i'd rather you not use those adjectives.)

Dodsy

Well, I'll respect that :)

Heather what math do you like?

Balarka Sen

Thanks. I think I am far from a genius. I know more than the other high school math enthusiasts here, but they are really very good at solving problems. I value that more than knowledge.

currently trying to get better at that to be honest

heather

@Dodsy well, I'm not very advanced

I am trying to more rigorously learn calculus, and I also dabble around - abstract algebra is pretty cool; I've got Durbin's book.

I love linear algebra, though I can't say I know it, but it has many applications and has such a beautiful geometric interpretation.

@BalarkaSen is far beyond me.

Dodsy

@BalarkaSen news.nationalpost.com/news/canada/…

I didn't even know he had a show.

heather

00:06

@Dair I responded to your comment

thank you, you brought up a good point - I completely missed that.

Balarka Sen

he does, but i haven't seen it. we just talked about a question on MSE he asked

and related things

Dodsy

Oh cool.

@heather That's cool, I wish I had that enthusiasm in middle school.

Zach's and Balarka's mathematical ability is well beyond mine, and I am 23 years old.

heather

it's never too late to learn =)

Balarka Sen

I agree.

Dodsy

@BalarkaSen Do you know of any numbers where the square root is equal to the sum of the digits?

Balarka Sen

00:09

Hmm!

Dair

assuming that math knowledge is a total ordering.

heather

@Dodsy, ooh, interesting question...

I bet you could write a program to solve that...

Balarka Sen

81 works, right?

heather

indeed.

Dodsy

OH YES

That was quick.

Just by brute?

Balarka Sen

00:10

Yeah. I think there should be more examples.

I'll let heather write a program

heather

you could argue that 1 works

Balarka Sen

yeah but meh

heather

yeah =P

Dodsy

hmm

I am not sure if there would be anymore.

Write the program heather.

For instance a 3 digit number the highest it could possibly count is 27

Balarka Sen

Good point actually.

Hmm. If I have a number of n digits, what the tentative size of it's square root? n/2 digits?

Right, because sqrt(10^n) = 10^(n/2)

Dodsy

00:13

that makes sense yes.

Balarka Sen

Maximum of sum of digits of an n digit number is 9n, like you said

Dodsy

right.

So to check it

heather

hmm, my program's kind of a brute force sort of program, and it's not coming up with anything other than 1 and 81

Balarka Sen

9n is very very small compared to 10^(n/2), which is the tentative size of it's square root

Dodsy

all you need to do is do 9n^2

heather

00:14

^that?

Dair

@heather Not gonna lie, the papers algorithm is kind of weird. I wouldn't have written it that way, and I'm not sure if there is something small that I don't understand about it. Your bottom code looks mostly correct except for some plurality issues, but I don't feel confident enough to answer it.

Dodsy

Heather 81 is the only number I believe.

Balarka Sen

@Dodsy Why n^2?

Dodsy

so 9 multiplied by the number of digits squared

heather

@Dair it is, I'm having a lot of trouble with it - do you happen to have a resource where I could learn to calculate/prove an algorithm's complexity?

Dodsy

00:15

(9n)^2

heather

also, what do you mean by plurality issues?

Dodsy

sorry forgot the brackets

Balarka Sen

Well, 9n = sum of digits. I wouldn't square that.

Dodsy

and if it's smaller than the number then it doesn't work.

Dair

You compute each element and then remove one element.

Dodsy

00:16

Basically it proves that 81 is the only number

Balarka Sen

But I think I sketched what you had in mind; sum of digits = 9n << 10^(n/2) = square root.

Yeah.

Dodsy

Yes you explained it best Balarka :)

It was an interesting thought though!

Balarka Sen

I liked the question

@Dodsy There's actually a lot of fun questions like this. Can you give an example of a number which is equal to the sum of factorial of it's digits?

(There's a 3 digit number)

Dodsy

hmmmmmmmmmm

heather

@Dair but only while the elements can be computed - once it reaches an element that can't, it removes that element and restarts

Daminark

00:19

@Dodsy starred for r/nocontext

Dodsy

haha

heather

@BalarkaSen, with that problem I realized that because the sum of the digits will be a whole number, you don't even have to square root - you just check if the sum of the digits is equal to the number itself.

Dodsy

okay so...

heather

the maximum total any number can have is 9n, where n is the number of digits.

so there's a limit on the value the total can have...

Dair

oh so you say while it is computable... most languages don't work that way but I guess pseudocode can be however you want it to be. As for computational complexity, I don't have a good reference. Might want to check cs.stackexchange

Daminark

00:21

@EricSilva Yo

Balarka Sen

Right. Usually the max total of digit is going to be very small compared to the square root.

heather

@Dair while there exists a real solution, I guess? I think you can code so that works...

Dodsy

254

no.

Balarka Sen

For n = 3 you get max total 27 like Dodsy said and square root is already of order sqrt(1000) = 31.blah

@Dodsy ooh close

heather

oh, wait...ack, i have no clue what i'm doing.

Dodsy

00:22

144

heather

(with the number problem)

Dair

@heather You can, but I'm not sure the pseudocode translates as easy as you think it does to python.

Balarka Sen

@Dodsy Did you mistype a digit?

Dodsy

gah

Dair

anyway I need to go for now. gl @heather

heather

00:24

@Dair, okay, thank you.

Dodsy

Okay I need help Balarka.

How do you do it.

This is what I know

number must be less than 7.

Balarka Sen

Both your answers were pretty close. What's the least number n such that n! is 3 digits?

Dodsy

5

which is 120.

Balarka Sen

Right. so you know 5 is going to be a digit of your number. can you amalgamate that with your other almost-answers?

Dodsy

well 5! + 4! = 144

Balarka Sen

00:28

yep yep yep

Dodsy

and 1! = 1

so it's 145

I am so stupid.

Balarka Sen

Great, thanks!

Dodsy

It was right there

I was so close.

Balarka Sen

Nah you were quite good

Dodsy

I kept using 5 and not thinking to put a 5 in there haha.

Balarka Sen

00:30

There's actually a bigger number. I think 5 digits?

Let's look it up on OEIS

Dodsy

Is there an infinite amount?

Balarka Sen

No, I think that's it. There's no more other than that 5 digit thing

Dodsy

Ah I see.

Balarka Sen

oeis.org/A014080

so it's 40585

Dodsy

Oh cool!

brb Balarka.

Balarka Sen

00:33

see ya

Ted Shifrin

00:45

Balarka, are you still not unsleeping?

Balarka Sen

I slept from 5PM to 1AM yesterday; my sleep schedule is in shambles

heather

um.

Ted Shifrin

you're totally a shamble.

heather

I feel bad for your Circadian rythyms

Ted Shifrin

he's a Circadian grasshopper.

Oh, never mind, that's a cicada.

heather

00:47

lol

Balarka Sen

Good pun nonetheless. Demonark-style.

Ted Shifrin

You might get ignored for less of an insult than that, Balarka.

Balarka Sen

:P

Daminark

I'm honored

Simply Beautiful Art

A new interesting pattern to $i↑↑n$ that looks cool if anyone is interested. The graphs are pretty cool looking.

Ted Shifrin

00:59

ponders dishonoring Demonark

heather

@SimplyBeautifulArt they are pretty cool =)

Simply Beautiful Art

@heather Fancy seeing you here

@TedShifrin D:

heather

@SimplyBeautifulArt rather strange, I know =)

Simply Beautiful Art

@heather Wait till you see $(-i)↑↑x$. It looks even crazier. Though I have to go to bed right now :(

heather

aw, bed interfering with mathematics?

=/

have a good night, anyway.

Simply Beautiful Art

01:04

Yeah =/

I'll try

Ted Shifrin

Wait 'til you're Simply's age, @heather.

Simply Beautiful Art

Hey look, its @EricStucky. Fancy seeing you here

@TedShifrin :-(

heather

@TedShifrin why, i thought @SimplyBeautifulArt wasn't that much older...?

Simply Beautiful Art

I'm 17

heather

eh, 2.5 years

okay, yeah, that's a ways from now.

=P

Simply Beautiful Art

01:06

@heather Is high school next year for you?

Balarka Sen

"Given the existence as uttered forth in the public works of Puncher and Wattmann of a personal God quaquaquaqua with white beard quaquaquaqua outside time without extension who from the heights of divine apathia divine athambia divine aphasia loves us dearly with some exceptions for reasons unknown but time will tell"

heather

@SimplyBeautifulArt yep

Balarka Sen

why do I have this stuck in my head

Ted Shifrin

go to bed, @Balarka.

Simply Beautiful Art

@heather :-) Hope you'll like it

heather

01:07

@SimplyBeautifulArt i'm more than a bit nervous

Simply Beautiful Art

@heather what for?

heather

i vacillate between the situation that i'll be very disappointed and it'll still be boring, and the situation that it'll be really hard and I'll fail

Simply Beautiful Art

You won't fail

:-)

Ted Shifrin

you'll do fine, heather

heather

thank you, but it worries me

everyone says honors english = tons of homework

Simply Beautiful Art

01:08

lol

Ted Shifrin

I remember being told I shouldn't take 4 honors courses. They were full of it.

Simply Beautiful Art

tons of homework not equal fail

It just means that its more work than you are used to

heather

yeah, but i have a bunch of extracurriculars i want to do - robotics, mock trial, quiz bowl...

@SimplyBeautifulArt yeah well =P

Simply Beautiful Art

Trust me, I have to write 10 page essays sometimes

heather

i actually don't really mind essays

Simply Beautiful Art

01:10

._. Can you do my homework then?

heather

um. i'd say yes but i think that's academic dishonesty =P

you know what's really terrible? I have to take health again next year. health.

::shivers::

Ted Shifrin

you don't want to be unhealthy

heather

i mean, i like the actual thing, just not the class

the class is awkward

Ted Shifrin

what is "the actual thing"?

heather

i had to take health this year too...we're in that unit right now.

Simply Beautiful Art

01:12

@heather sh...

heather

@TedShifrin being healthy, like in real life.

Ted Shifrin

well, that's not a bad thing!

Simply Beautiful Art

@TedShifrin You don't get healthier by taking health class in my opinion

heather

^that's what i mean.

Ted Shifrin

well, being educated helps

Simply Beautiful Art

01:12

Meh... I'd rather do real exercise and stuff

Ted Shifrin

but, no, I had two heart surgeries and a cancer surgery, and it wasn't due to my ignorance ...

yes, real exercise is super important

heather

also, the unit where you talk about stuff in a co-ed class is awkward...

@SimplyBeautifulArt yeah, I actually kind of like P.E.

it's a nice break.

Simply Beautiful Art

@heather I liked dodgeball

@TedShifrin Oh noo....

heather

@SimplyBeautifulArt haha, yeah. i like playing soccer =)

@TedShifrin i'm so sorry

Ted Shifrin

I thought dodgeball was super American. @Simply, you're not in the US, are you?

no need to be sorry, @heather. I was just stating facts. I'm doing quite well.

Simply Beautiful Art

01:14

@TedShifrin I am. What made you think otherwise?

Ted Shifrin

Oh.

Simply Beautiful Art

Also enjoyed flag football.

Ted Shifrin

Isn't 9:15 early for bedtime on a Friday?

heather

have you ever done rock climbing @Simply?

Simply Beautiful Art

@heather nah... I'm scared of heights

@TedShifrin sh!

heather

01:15

oh.

Ted Shifrin

I'm too much of a klutz for that, heather.

Simply Beautiful Art

^

heather

@TedShifrin i'm a klutz too, and it's still really fun - I'm on a team that trains and competes

Ted Shifrin

Good for you!! I still love volleyball and tennis ... although my back betrays me now.

Semiclassical

hi chat

heather

01:17

hello

Simply Beautiful Art

Hi

Ted Shifrin

oh no, it's Semiclassic.

5

Semiclassical

lol

I could really horrify you and talk about a research question :P

(i say that, but i'm not sure i'd really want to anyways.)

Simply Beautiful Art

Or we can talk about my weird $i↑↑x$ graphs

That actually look cool :P

Ted Shifrin

You're asleep, @Simply.

Simply Beautiful Art

01:19

@TedShifrin SH!!!!

But it does look cool

Ted Shifrin

WTH is Eric doing back?

Simply Beautiful Art

lol

@TedShifrin Sleep is for the weak.

Ted Shifrin

Or for the weekends.

heather

anyone in here know how to calculate algorithmic complexity?

Ted Shifrin

Not I.

heather

01:22

specifically, I want to know a. whether or not this algorithm is polynomial time and b. whether or not this algorithm is more or less efficient than another.

Simply Beautiful Art

@TedShifrin tru tru, but not at night, sleep through the morning and wake up in the afternoon.

heather

the algorithm in question being the third here, compared to the first on the same question

Simply Beautiful Art

@heather I know of it a bit, but not enough to help at all.

Or maybe enough to help

heather

::hopes it is enough to help::

Simply Beautiful Art

but not enough to understand what your code is doing...

or... I shall go to bed

g'night all!

heather

01:23

ah, okay. good night (for the second time =P)

Ted Shifrin

keeping mum

Simply Beautiful Art

=P

WHOA! I KNOW THIS @ZAID PERSON!

:O

Ted Shifrin

night, Simply.

Zaid Alyafeai

Heyy

@SimplyBeautifulArt, how are you doing ?

Simply Beautiful Art

@ZaidAlyafeai Supposedly sleeping, you?

Zaid Alyafeai

01:27

@SimplyBeautifulArt, Watching Cavs versus Celtics play offs match

Simply Beautiful Art

@ZaidAlyafeai Perhaps you can help with this

And g'night for real

heather

@SimplyBeautifulArt g'night for the third time

Daminark

Just got back @Ted and :O don't dishonor me

Ted Shifrin

It's hard not to dishonor you, Demonark.

Daminark

Impossible!

Dodsy

01:41

daminark*

Demonark would be cooler hough

Daminark

That's Ted's nickname for me

Dodsy

Aren't I just out of the loop

Daminark

So it seems :P

Dodsy

@TedShifrin can you view site analytics?

I thought it'd show my flair.

Now I see that it looks sketchy

AKIVAAAAAAAAAAAAAAAAAAAAAAAA WEINBURGER

Daminark

01:57

Yo @Akiva Laci was explaining the integral solution to me earlier

He said it's basically the same as the checkboard one in a way

You could see it as a signed measure, with one color being positive and the other negative

So that's kind of why the integral works

Dodsy

imgur.com/a/HoxSq

"control of the mighty pickle"

Daminark I have a somewhat easy math question for you.

Which I thought up earlier.

Daminark

Let's hear it

Dodsy

Okay find a number whose digits add up to the number's square root, and prove that this is the only number with this attribute.

Dair

1

Daminark

1

Darn @Dair sniped

Dodsy

02:08

Other than 1...

MickLH

Can the number be the average of some non-terminating decimal expansion?

Daminark

81

Dair

didn't you do a variant of this problem earlier today?

Dodsy

Yessssssss

Prove that it's the only number Daminark

other than 1.

Daminark

Well, you can manually handle 2 digit numbers

And for 3 digits and above, I think you'd use something like making all the digits 9 and proving it's never enough?

Dodsy

02:11

Yeah that's it.

I guess it was too easy for you.

Daminark

I mean it's a good problem for sure

Balarka Sen

The one pattern that always unnerved me was 3^2+4^2=5^2, 3^3+4^3+5^3=6^3 but that's it.

Dair

Why does it unnerve you?

Dodsy

3,4=5

3,4,5=6

Just the ascending nature.

Balarka Sen

3^4+4^4+5^4+6^4 $\neq$ 7^4

@Dair It's strange, innit?

Dodsy

02:14

Hm.

Dair

i misread the first time.

I thought you said $3^2 + 4^2 = 5^2$ and $3^3 + 4^3 = 5^3$

actually wait

Balarka Sen

#FLTrekd

Dair

that doesn't make any sense lmao. i'm dumb

classic fermat

Balarka Sen

It's a coincidence, that pattern, I'm sure. (Also known as "strong law of small numbers") But a weird one.

Dair

number theory patterns are always fun to look at.

Balarka Sen

02:18

true

Dair

and i can explain them to non-mathy friends. :)

(well in a lot of cases)

it's always fun to quote project euler solutions to my friends and at the end be like: "The more you know, the better prepared you'll be for tomorrows problems"

heather

@Dair, fyi, I significantly updated the question

Dair

@heather Ok, I'll take a look, but I'm not sure how much I can help.

heather

@Dair, you don't need to take a look; I just know you were interested earlier so I thought I'd let you know.

Dair

it's rather weird they use a goto at all, not just in python, but in every language it is considered a bad practice in general

heather

02:21

but if you do, thanks

Dair

ah ok.

heather

@Dair yeah, I've received several comments to that effect and also to the general effect that the paper was very confusing

(in the way it was written)

Excalibur42

Man I really can't deal with sequences of sequences questions

can anyone explain to me how the comment on my post here (math.stackexchange.com/questions/2287327/…) gives total boundedness of that set?

heather

03:02

good night everyone

BAYMAX

04:01

$(1+a+b+c).(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}) = 16$ then $a+b+c = ?$

hints?

arctic tern

do you have an integer solution?

BAYMAX

yes

I see that $a=b=c=1$

satisfy but how to show that this is the only triple?

like the sum $a+b+c$ is fixed!

Sorry I forgot to mention $a,b,c$ are positive real numbers.

Daminark

04:22

Hey everyone!

BAYMAX

hey@Daminark

any chilling soundtracks anyone?

Daminark

9 Hours, 9 Persons, 9 Doors - Chill and Rigor

►

@Balarka So I think I may have a way of proving that if $p\equiv_4 1$, then it is not Gaussian, which does not use the Fermat squares thing.

If $p$ is prime, you have a primitive root mod $p$

Call it $r$

Now if $r^{p-2} \equiv_p p -1 \equiv_4 0$

Wait merp not sure how to continue

So $p - 1 = 4n$, and $r^{p-2} - 4n = pk$

Daminark

04:48

Wait I misread the existence of primitive roots

OK so we know we have some $k$ such that $r^k \equiv_p p - 1$

And we know by assumption that $p - 1 \equiv_4 0$

BAYMAX

$f:[0,1] \rightarrow R$ , non-negative with $\int_{0}^{1}f^{n} dx \rightarrow 2$ as $n \rightarrow \infty$ , is this possible ?

Daminark

Is $f^n$ a power or is it composition?

BAYMAX

Power!

Daminark

Then I think $f(x) = \sqrt{2}$ might work? If I remember correctly

Also hey @Dair

Dair

Hey @Daminark

Daminark

04:57

Wait a second I think this might amount to proving the squares thingy

Dair

$f(x)^4 = \sqrt{2}^4 = 2(2) = 4$. But $\int_0^1 f^4(x) dx = 4$...

Daminark

Wait nevermind I was thinking of something else, $\sqrt{2}^{\sqrt{2}}$ and iterating that

Nevermind

Dair

I feel like something like a bump function should work...

Daminark

Yeah a constant could never work, either it explodes or it tanks to 0

Right now I'm not wearing my analysis cap :P

But yeah so assume $x^2 \equiv_p -1$, then $x^4 \equiv_p 1$

BAYMAX

bump function? @Dair

Transcript for

$Mathematics$ Mathematics