Just a Simple and Innocent Question

Is it true that every even number greater than 2 can be written as a sum of two primes?.. 😉

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Published in: on Wednesday, July 16, 2008 at 5:30 pm  Comments (18)  
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Jack Edmonds – One of the Last Century’s Greatest Mathematicians

I am turning to all of you, my dear readers, with this announcement/application.

 

I have to tell about the great American mathematician Jack Edmonds, and his works in my PhD course after a couple of weeks, and I am now collection all possible information regarding to this man. I have read his most famous article `Paths, Trees, and Flowers`, I have found some citations other people said about him, I have studied some problems connected with his work, etc.

 

What am I asking – can you tell me something more about Jack Edmonds, like some facts from his biography, some interesting stories about him or that he has told, about his personality, etc. It surprised me how little information I could find about his life, including the title of his dissertation – I just couldn’t find it.. So, can you point me, where can I find something more about him? The most valuable would be to get to know, what impact his most famous work left on the subsequent mathematicians – either your own opinions or citations of some famous mathematicians or other people..

 

P.S. Some photos of Jack Edmonds would also be appreciated.

 

The Curiosity About Prime Numbers

Everybody knows prime numbers. Most of us like prime numbers. Many of us love prime numbers. Some of us have noticed some curious things about prime number. Let me mention some of them:

2 – the curious thing about the prime number 2 is obvious – it is the smallest prime
3 – the smallest odd prime
11 – the smallest 2-digital prime
16 – the smallest number that is the fourth degree of some prime

Etc..

So, if we keep our mind busy thinking about curiosity of small integers, we will certainly be able to think of something interesting (related to primes) about every positive integer. But how far can we go? OK, we can think curious things about integers smaller than 10 – I’ll assume everyone can do that. Well, maybe we can assign a curio to every integer smaller than 100 as well. I guess, not everyone could do it so well this time, but, in fact, it could be possible for some of us. But what’s next? Can we go further till 1000? Can somebody tell me, for example, what is the curiosity about the integer 857 (it must be related to prime numbers)? Not instantly, I guess.. Therefore I was amazed by the theorem I found in this site:

Theorem. Every positive integer n has an associated prime curio.

And do you know what? There exists a proof for this theorem as well:

Proof: Let S be the set of positive integers for which there is no associated prime curiosity.  If S is empty, then we are done.  So suppose, for proof by contradiction, that S is not empty.  By the well-ordering principle S has a least element, call it n.  Then n is the least positive integer for which there is no associated prime curio.  But our last statement is a prime curio for n, a contradiction showing S does not have a least element and completing the proof.

Well, I don’t know about you, my dear reader, but I found this proof very exiting. It actually means we could think of a prime curiosity for every positive integer, if only we thought hard enough and long enough. Very beautiful result, indeed!

What Does This Function Do?

1. Please, give your answers in comments – what does the function written below do:

a)      recognizes prime numbers

b)      recognizes prime numbers in interval [100;999]

c)      neither a) nor b) is correct

d)      other variant (please, specify)

2. And the second question is – is this function a correct solution if the task is as follows: `Please, write a function that recognizes all the prime numbers consisting of exactly 3 digits`:

a) yes

b) no

The function (in C++):

bool f(int x) {

 for (int y=2;y<x;y++)

  if (x%y==0) return false;

 return true;

}

P.S. I wish all of you wrote your thoughts in order to make a student of mine understand the essence of the thing!

Music – a Special Form of Mathematics

Music is the pleasure the human mind experiences from counting without being aware that it is counting

/Gottfried Leibniz/

I read it here and I found it to be true!

Published in: on Thursday, January 17, 2008 at 12:03 pm  Comments (2)  
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Random Number Generator – As Easy As 01, 10, 11..

Someone told me it is not a trivial task to make a good function generating a random number. What a fool I was to believe him! It turns out you can write down the function very easily. For example, in C++:

int getRandomNumber() {

   return 3; // proven to be random – obtained by throwing a dice

}

Not too complicated function, eh?

Published in: on Thursday, January 10, 2008 at 6:50 pm  Comments (2)  
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Blog Stats Is an Interpolation of What?

When looking at the diagram of `Dashboard à Blog Stats` in my WordPress, I can’t stop wondering about this curve – it must be an interpolation of some function. How could this function look like? What would be other arguments of this function except for the date (what are factors that affect the value of this function)? Can anybody define it precisely? Or not so precisely?

Published in: on Thursday, January 10, 2008 at 2:43 pm  Comments (7)  
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A Math Game

I attended the seminar for PhD students on Thursday and we played one game there. It turned out to be very interesting, so I want to share this with you, my dear reader.

So the rules are as follows:

The team consists of seven people, all labeled with one label – either red or green. The manager of the game do the labeling based on random label generator, that it, the possibility to get the green (or red) label for every person is equal to 0,5. Before the labeling process people are allowed to speak to each other and to agree on some strategy of the game. After the labeling is done, no communication is allowed, except everyone can see labels of other persons. The goal of the game is to guess the color of his label for some of the participants of the team. The guessing happens when the manager of the game asks it all the persons one after another. Every person can either pass (don’t guess his color), or guess his color, but it is not allowed for every person of the team to pass. When a person is surveyed and he makes a guess, the game ends – the team wins if the person guesses correctly, and loses if the guess is wrong.

The actual process can be seen like this:

  1. all the persons of the team is allowed to talk whatever they want for some time;

  2. the time is up, and all the persons settle in a row or whatever;

  3. the manager of the game goes to each person, chooses a label (remember – the probability of both colors is the same) and sticks it to forehead of that particular person (in the way person does not see the label);

  4.  when all the persons are labeled, everyone is allowed to look around to see the labels of other persons;

  5. when the looking is done, the manager of the game starts asking the question `What’s your color?` to each of the participants in an arbitrary order;

  6. person can say either `I pass` or name one of two possible colors – red or green (except for the last person who is forced to guess the color if it comes to him);

  7. if some person says `I pass`, the manager asks the same question to the next person;

  8. if some person guesses the color, the game is over and the team wins or loses depending on the correctness of the guess.

 So, you task is now to come up with a strategy that gives the biggest probability for a team to win. And it would be nice, of course, to motivate this strategy and/or to calculate the winning probability of your strategy. Who offers the best strategy (the biggest winning probability) wins (sorry, no prices though).

Published in: on Saturday, December 8, 2007 at 3:07 am  Comments (1)  
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Interesting Research Field

I can’t find the site right now, but I remember one scientist telling his research area in one short sentence:

My research is about what we can’t do with computers we don’t have.

He was, of course, working with quantum computing.. Can someone tell me the name of that scientist?

Discount or Surcharge?

It’s really irritating how people use negative integers (or real numbers) in place and out of place. I feel they don’t even know the difference between, let’s say 10% and -10% of the value of some product. It can be very amusing – when it needs to be told how warm or cold is it outside, nobody thinks – 10 degrees are the same as just 10 degrees 😉 However, when it comes to prices and discounts of prices, everyone goes mad:

Q: How much is that TV set?

A: It’s usually, $ 200.00, but now it has a discount of -10%..

So if it has a discount, it means you have to subtract the amount of the discount from the original value to get the new value, right? So, 200-(-10% of 200) = 200-(-0.1*200) = 200-(-20) = 200+20 = 220. Hip, hip, hurray, the new price after applying the discount is $ 220.00! J And what’s even funnier, it’s not just about the average person inhabiting the world of ours, it’s even more about all the companies and enterprises putting those “Sale! All the merchandise’s having a discount of -20%” labels in their windows. Could somebody, please, teach them a bit of math? You can say either of the following, but, please, don’t mix them together:

  1. all the merchandise has a discount of 20%;
  2. the new value of all the merchandise is -20%

P.S. This is my 17th post in this blog, so let’s remember that thought about 17 being an evil number! 😉

Published in: on Monday, November 26, 2007 at 1:16 am  Comments (4)  
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