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A quick look at the wikipedia entry on mathematical constants suggests that the most important fundamental constants all live in the immediate neighborhood of the first few positive integers. Is there some kind of normalization going on, or some other reasonable explanation for why we have only identified interesting small constants?


EDIT: I may have been too strong in some of my language, or unclear in my examples. The most "important" or "interesting" constants are certainly debatable. Moreover, there are many important and interesting very large numbers. Therefor I would like to make two revisions.

First, to give a clearer idea of the numbers I had in mind, please consider such examples as $\pi$, $e$, the golden ratio, the Euler–Mascheroni constant, the Feigenbaum constants, the twin prime constant, etc. Obviously numbers like $0$, $1$, $\sqrt2$, $...$, while on the wikipedia list, are in some sense "too fundamental" for consideration.

This leads me to my second revision, which is that the constants I am trying to describe are (or appear to be) irrational. Perhaps this is a clue to what makes them interesting. At the very least, it leads me to believe that large integer counterexamples do not satisfy the question as I had intended.

Finally, if I could choose a better word to describe such numbers, it might be "auspicious" rather than interesting or important. But I don't really know if that's any better or worse.

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    A mole isn't a math constant, it's a unit conversion number for physical quantities.2012-03-16
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    There's [Graham's number](http://en.wikipedia.org/wiki/Graham%27s_Number)...2012-03-16
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    Maybe it is a result of them being easier to discover and work with. The order of the monster group may be just as fundamental, but less has been discovered about it.2012-03-16
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    To a mathematician, every constant should be interesting :P2012-03-16
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    Dare I suggest $\sqrt{163}$ as a counterexample?2012-03-16
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    @ArturoMagidin, Graham's Constant is only interesting because it is large, so it doesn't really count - it doesn't have a singular play like $\pi,e,$ etc. Obviously, it depends on what we mean by "interesting," but I think the poster was referring to a deeper sense of interesting2012-03-16
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    Theorem: Every positive integer is interesting. Proof: Suppose there exists uninteresting positive integers. Then there must be a smallest uninteresting positive integer. Since being the smallest uninteresting positive integer is interesting, we have a contradiction.2012-03-16
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    You might read [this](http://math.stackexchange.com/q/98205/19341)...2012-03-16
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    Sometimes I think there is something funky going on with $\pi$ and $e$ and 2 and 3 -- the only consecutive primes, but every time I get a whiff of whatever connection there might be, it's too hard to translate it into something useful.2012-03-18
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    I think "interesting" is rather subjective. You should make your question more specific. I suggested Graham's number as Arturo.2012-03-18
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    @PeterT.off, fair enough, "interesting" may not have been the best adjective, nor qualifying the set in question as "all" the constants. There are certainly interesting large numbers. Perhaps "common" or "useful" are better descriptors of what I'm after. Although at the end of the day I will admit there is probably some unavoidable subjectivity here. As yet, however, I am not convinced the question is unanswerable...2012-03-18
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    @rar, are you suggesting that the smaller constants are simply easier to calculate?2012-03-18
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    I agree with @rar's opinion. The Wikipedia table is a list of known constants. It's possible that we are going to see many large numbers in that table 20 years from now. We mathematicians just need to work harder to discover them. That's all.2012-03-18
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    @GregL In part. Until the last 30-40 years dealing with specific large numbers was extremely difficult computationally, so there really has not been very much time, mathematically speaking, in which it is likely anyone would have run into large, fundamental constants, and if they did, it would have been difficult to develop much theory surrounding them. The constants we think of as "fundamental" are simply the ones for which we have the benefit of 2000 years of study and discovery. I would guess 2000 years from now there will be many "large" constants considered fundamental.2012-03-18
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    Size isn't everything.2012-03-19
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    But what is the smallest non-interesting constant?2012-03-20
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    This question is being discussed on [reddit.com/r/askscience](http://www.reddit.com/r/askscience/comments/1rlfdu/is_there_a_reason_why_most_of_the_important/). There are some interesting thoughts there that I don't see here (and vice versa), so I thought people might like to know.2013-11-28

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