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I'm sure most of you have heard of the argument that all numbers are interesting: If any uninteresting numbers exist, there must be a "smallest" uninteresting number, which would make it interesting – a contradiction. But I don't buy that. I would like to find at least 2 interesting things about every number below 20. I realize that this is an odd request, but I am making a presentation and would appreciate your input. Here is what I have so far:

0- It is one of the fundamental mathematical constants. Nuff said.

1- It is one of the fundamental mathematical constants. Nuff said.

2- Only even prime; a^n + b^n = c^n only works for n<2;

3- First odd prime; It is a Heegner number.

4- Number of quadrants for the Cartesian Plane; It is a Smith Number.

That's all I have so far. Any input would be appreciated. Thanks.

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closed as not constructive by Aryabhata, Rasmus, Zev Chonoles, t.b., yunone May 21 '11 at 0:20

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I think you mean $n\le2$. –  Arpon May 20 '11 at 23:41
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5 is the first number not on your list. Also it is the only prime that is divisible by 5. –  Myself May 20 '11 at 23:59
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I suggest that you have a look at en.wikipedia.org/n_(number). e.g. 5, 6, 7, 8, etc. This isn't mathematics, this is numerology, to avoid a word that starts with s and ends with illy. I'd be inclined to add plain, but then I'd probably get this comment flagged. –  t.b. May 21 '11 at 0:14
    
There are a few books on the topic. Le Lionnais, Les Nombres Remarquables; Joe Roberts, Lure Of The Integers; finally, there's The Penguin Book Of Curious and Interesting Numbers by David Wells. –  Gerry Myerson May 21 '11 at 0:32
    
If you think that no numbers are interesting for any other reason, than 1 is the smallest positive integer that is uninteresting, 2 is the smallest positive integer among the uninteresting ones that is not the smallest uninteresting integer, 3 is the smallest positive integer among the uninteresting ones that is neither the smallest uninteresting integer, nor the smallest positive integer among the uninteresting ones that is not the smallest uninteresting integer, and so on. ;-) –  Luboš Motl May 21 '11 at 12:55
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4 Answers 4

up vote 0 down vote accepted

4 - minimum number of distinct colors required to color any planar map such that no two adjacent regions have the same color.

5 - the smallest number of pieces into which a 3-dimensional ball can be divided and rearranged into two identical copies of the original ball, as per the Banach-Tarski paradox.

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You seem to be one of the few people that understands where I'm coming from. –  Hautdesert May 26 '11 at 2:42
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It only has one, but goes much farther than 20: What's Special About This Number?. I have a bookful at home as well. This one only goes up to 9, but has a whole chapter for each.

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Wow... you managed to answer without being "subjective and argumentative"! (+!) –  The Chaz 2.0 May 21 '11 at 4:55
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3: First positive number $n$ such that there is no $n$-dimensional skew field over the reals.

I've got a sinking feeling that most answers here will be of the form '$n+1$: first number > $n$ such that ...' so I've flagged this question for moderator attention - but I may as well get this fundamental example in :-)

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These two sites may help for your presentation:

http://www.numbergossip.com/5

http://www.wolframalpha.com/input/?i=5

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