Archive for May, 2010

The Number Game (3)

May 30, 2010

Got my laptop back, so quasisemisortofseriousposts shall will be going to be written again next week. For now, in honour of last week’s Towel Day, the number of the week is the expected number of throws of a die until two sixes show up successively for the first time. Happy hunting. Although it’s too obvious. Whatever.

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The Number Game (2)

May 26, 2010

My laptop and I haven’t been together for some days now, so I couldn’t update you on my interesting life. But apart from being sleep deprived, cycling the elfstedentocht and (probably) not being able to pass a lineair algebra exam, because of sleep deprivation and pain everywhere, not a lot happened. So, basically, all I have to tell you is:

2^{45} * 3^{13} * 5^6 * 7^2 * 11 * 13 * 17 * 19 * 23 * 31 * 47

Good luck!

Nailing my life

May 21, 2010

Some days are just plain motherfucking awesomeness. Like yee-esterday. And by yesterday, I mean the past 30 hours, because I kinda screwed up my sleep rhythm. Let me tell the stories.

One of the first things I noticed when I woke up (approx 8 pm on May 19th), was an e-mail from mister Nicely, telling me that my proof was correct. That is, I showed that for every integer k > 1, an integer x exists, such that: \displaystyle \frac{x}{\pi(x)} = k, where \pi(x) is the prime counting function. Since I proved that fact (it definitely does not earn the title ‘theorem’) about a year ago, I was quite surprised to see it as a conjecture of Golomb on Nicely’s site. So I e-mailed my proof, and it got posted. While this could easily be considered a trivial story, and that claim has merits, I’m actually quite happy about it. Because I not only proved the above fact, I also proved my existence! Furthermore, I started thinking about the math in question some more, and I came up with the following conjecture, which, to me, seems certainly true: for every positive integer m, there is a positive integer k, such that there are at least m different positive integers x, for which \displaystyle \frac{x}{\pi(x)} = k. Maybe even the following is true: for every positive integer m, there are infintely many positive integers k, such that there are exactly m different positive integers x, for which \displaystyle \frac{x}{\pi(x)} = k. The second one seems quite hard (maybe the case m = 1 is doable), but I thought about my first question some time, and I think it should be attackable. I (think I) am able to proof the conjecture, under the assumption of Hardy-Littlewood’s prime-k-tuple-conjecture, if we drop the condition that k is an integer, but only want it to be rational. Although I must add that I haven’t written down the details yet. I might put my thoughts on it here. Butttt, obviously not now, because I have to tell you the rest of mah story!

So, by (what felt like) morning, I was already world-famous. Then my calculus-teacher e-mailed me. And that event was something I had been dreading for like a month. Because I asked him to e-mail my grade on the exam and I wasn’t quite sure about it. Because, honestly, I HATE CALCULUS. And I’m obviously talking about the way it’s taught and the fact that you won’t be allowed to come back ever again if you use your intuition for something. I mean it’s good for students that have never seen proofs before and shit, but come on, be less trivial, puh-lease. Honestly, I might roundhousekick the next person who asks me to prove that \displaystyle \lim_{x\rightarrow 0} x = 0 straight in the face. Or, somewhat more generally I guess, the next person who uses the words ‘epsilon’ and ‘delta’ in one sentence. And this hot, raging anger I felt inside me, made it almost impossible to open my book to study without hysterically crying. Add this to the fact that I usually only get part marks  (I mean, what’s up with that?), when I answer a question like ‘prove *something trivial*’, with ‘that’s trivial’ and it becomes clear why I was scared that I failed my exam like a little bitch. As it turns out: I PASSED. Go suck on that you Cauchy!

And these 2 major steps towards eternal glory made me

1) call my lovely little sister for like 40 minutes
2) able to help out a friend (which wasn’t even math-related)
3) impulsively (though wisely) decide to buy a new phone
4) notice that it was like the most beautifullest weather ever today
5) write a post that has (ugh, too geeky) \lfloor 200*\pi \rfloor words in it

Congratulations!

May 18, 2010

While busy studying (click here if you want to know how that’s going), I almost forgot that my blog turns 1 week today! So, with all my love: Congratulations!

When will we know primes?

May 18, 2010

Because prime numbers fascinate me, I was thinking about some conjectures about them. And some of them just seem hopeless to me. To check if you agree with me, I came up with 3 polls, about the following conjectures:

Cramér’s conjecture
Abc conjecture
Bateman-Horn conjecture

I SUCK I SUCK I SUCK

May 17, 2010

Well, see title.

The Number Game (1)

May 15, 2010

It’s time for something geeky. I’ll post a number, you ‘guess’ why it’s cool. Obviously, this is, in essence, subjective, so you might question the existence of a wrong answer, but I’ll try to make it as clear as possible. So when I say ‘1729’ and you answer ‘that’s a cool number, because when you reverse its digits, it’s the zipcode of my aunt’, you’re just wrong ;). Because everybody knows that 1729 is the second taxi-cab number. Otherwise known as the Hardy-Ramanujan number. Alright, an easy one to start us off:

11011000001

I’ll post an interesting number once a week. Also, when you comment, could you please hide your comment if it contains (possible) information on the solution, so as not to spoil? A simple way to do this is to use the colour white; 

<font color=”white”> Your text should be here </font>

But there might be a more advanced way. Anyway, happy puzzling!

Help

May 15, 2010

Alright, fuzzy story, clue can be read below. So far I’ve done the following if I wanted to post something: Type the post in \LaTeX, copy/paste the post here, put $latex-signs everywhere to tell WordPress you’re using \LaTeX, and done. However, today I realised how wrong this is. Apparently, and this is something I should’ve known, but didn’t, WordPress isn’t able to work within environments that, in \LaTeX, say ‘change from normal text to mathematical text and do something beautiful (like aligning equations)’. Because when you want to use \LaTeX in WordPress you have to put a $latex-sign in front, like I said, but Wordpress then automatically thinks that you’re writing math, so it changes to math-text. So you can’t tell WordPress that you want to change to math-text, because it already assumed that! But that also implies that you can’t use environments that tell \LaTeX to go from normal to math text. Wow, that was even fuzzier that I thought it was going to be.

Long story short: I’m not able to display equations as neat as I want. In particular, the thing that tilts me the most is my inability to align equations. So I searched the forums, but no help. Then I started to try random stuff, didn’t help either. The best I was able to do, is something like the following:

\begin{pmatrix}a^2+b^2&=&c^2\\&=&d^2-e^2\\&=&(d+e)(d-e)\end{pmatrix}

which, in a very basic way, resembles something that, when I really want to see it, looks like it’s not insanely far from what I might want. But in any case, it just feels ridiculous to put equations in a matrix.

So: please please please help me. Either by telling me how to use the align-environment, without WordPress realising it, or by telling me that the above is, in fact, the best I can do, or by convincing me that it doesn’t really matter that much how math looks. Go!

EDIT: I found a way to do it, but that is like a hell. So if you have anything useful to say, please, still comment.

Clearing the PEN-Book

May 14, 2010

Just a quick post to say ‘Wieeeee’; today I solved the following problems, all from PEN;

1) Prove that for every n, the set of unit fractions contains a maximal arithmetic progression of length n. Maximal means that, if \displaystyle \frac{p}{q} is the common difference, \displaystyle \frac{1}{x_1} is the largest and \displaystyle \frac{1}{x_n} is smallest term of the arithmetic progression, then both \displaystyle \frac{1}{x_1} + \frac{p}{q} and \displaystyle \frac{1}{x_n} - \frac{p}{q} are not unit fractions

2) Prove that a 3*3 magical square cannot contain only Fibonacci numbers

3) Prove that for every n, there are 2 consecutive integers that are the product of n integers

Since my thought process, nor my proofs are interesting, I won’t give any details here. Although I must say that changing ‘3*3’ to ‘m*m’ in problem 2) and disallowing the use of induction in problem 3) makes those problems slightly more interesting. Even better: disallowing the use of dirichlet’s theorem on arithmetic progressions in problem 1). I have, so far, not been able to solve that one, although I didn’t think that hard about it yet and I might be indoctrinated, of course. I also tried to prove that there are 2 consecutive integers that are the product of n distinct primes for every n, which, to me, is a much more natural question to ask than problem 3). But I quickly realised that there is a reason that problem 3) is what it is, because my generalization trivially implies that there are infinitely many consecutive numbers with the same number of divisors. And this is known to be a hard problem and was, according to Wikipedia, only settled in 1984, so I just gave up on that. Maybe the following is ‘reasonably’ attackable: prove that there are 2 consecutive integers that are both the product of n distinct prime powers for every n. But I wasn’t able to solve that one either. If you have any interesting thoughts on anything I said here, don’t hesitate to comment. Anyway, wieeeee

To teach and learn

May 13, 2010

When you post your solution of a math problem somewhere, there are, very roughly speaking, 2 ways to present it:

1) Like writing a math paper; proving everything strictly, without appealing (too much) to intuition and usually written in a totally different order, than the way it was solved
2) Describing the whole road you took; starting with the problem, initial thoughts, ideas, fallbacks, vague notions etc.

Clearly something in between is possible too. Anyway, I think it’s quite obvious that I went with 2) when I blogged about the coin-flipping problems. And since way 1) of presenting math has a lot of advantages, you might wonder why I went with 2). I’ll tell you.

If someone else than me is actually trying to gain anything by this blog, chances are that he or she is not used to the way math is usually presented. Because, from my personal experience, when I talk to a non-mathematician about math and am, for example, explaining my proof of some theorem or my solution to a problem, the response I get the most is ‘alright, I get this and that when you explain it, but I would never be able to come up with it myself’. This could obviously be due to the fact that that person is just not as smart as I am. But I beg to differ. Well, of course, it’s usually trivially true that I am smarter than that person, but that’s not the issue; I don’t think that’s the reason. The reason that that person (ah, let’s just make that person a she), the reason that she is not able, or thinks she’s not able, to come up with a solution, is just because she never learned to think like a problem-solver. So, if there’s anyone out there that reads this blog and tries to learn something from it, I want to help her (yeah, it’s definitely a her) as much as I can. And I think the way to teach somebody to solve a problem, is to show her some possible lines of thought. Show her the things you thought when you thought about the problem. Show her the ways you tried to overcome certain obstacles on the way. Show her some problem-solving techniques. Some will work on some problems. Most won’t work on most problems. But that’s ok. Because when you give a non-mathematician a problem and ask what line of attack is the best at first sight, she usually doesn’t have a clue. And I think teaching someone how to get an idea, only a tiny little idea to start with, is all you have to do.

Alright, let me rephrase the above paragraph in a language that’s somewhat more egocentric;

Since I´m probably the only one that is actually trying to gain anything by this blog, I have to realise that I´m not used to the way math is usually presented. Because, from my personal experience, when I read about math and am, for example, trying to get someone´s proof of a theorem, or some guy´s solution to a problem, the thing I think the most is ´alright, I don´t get this, nor that, let alone that I will ever be able to come up with it myself.’ This could obviously be due to the fact that that person is just smarter than I am. But I beg to differ. Well, of course, it´s usually trivially true that that person is smarter than I am, but that´s not the issue; I don’t think that’s the reason. The reason that I am (ah, I wish I was a girl), the reason that I am not able, or think that I’m not able, to come up with a solution, is just because I never really learned how to think like a problem-solver. So, since I’m writing this blog for myself and I am going to try to learn something from it, I should try to help myself (nah, being a boy is just fine) as much as I can. And I think the way to teach myself how to solve a problem, is to write down some possible lines of thought. Show myself the things I thought, wanted to think, or should have thought, when I thought about the problem. Show myself the ways I could have tried to overcome certain obstacles on the way. Show myself some problem-solving techniques. Some will work on some problems. Most won’t work on most problems. But that’s ok. Because when I see a problem and ask myself what line of attack is the best at first sight, I usually don’t have a clue. And I think, teaching myself how to get an idea, only a tiny little idea to start with, is all I have to do.