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Are there some examples of mathematocal constants which are integer numbers. I know of one that is called Kaprekars constant but thats just a base 10 curiosity. Aret there some more important examples? perhaps in the fields of combinatorics or abstract algebra? Thanks. It would be optimal if it where 4 digits long.

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Integer and yet transcendental? Impossible! –  Marc van Leeuwen Oct 15 '12 at 14:04
    
I dont mean trascendental in the mathematical sense. I mean important. –  user4140 Oct 15 '12 at 14:09
    
This sounds like "soft list" - should this be community wiki? –  Gottfried Helms Oct 15 '12 at 14:24
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"It would be optimal if it were $4$ digits long". Huh? Optimal by which criterion? Are you looking for a PIN code you can remember? I've noted a lot of recent math papers mention the integer $2012$, that would fit the bill I guess. –  Marc van Leeuwen Oct 15 '12 at 16:24
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5 Answers

up vote 1 down vote accepted

Integer constants: what do you want?

Transcendent numbers?

  • $\pi$ ?
  • e (=exp(1))?

Is that really your question?


Ok, another try after your comment:

  • 11 - the first prime p such that the mersenne number $2^p - 1$ is not prime?
  • Graham's numbers?

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The last two arent integers so no. The first three ones are integers but they are not one single number. Wht i am looking for is a single integer which can be used to solve various problems. Or appears in several places. like pi.or mabye some important theorems in combinatorics or abstract algebra which state impossibilities for things larger than n (where n would be the constant im looking for. –  user4140 Oct 15 '12 at 13:50
    
hmm, you asked for "more transcendental examples"... ;-) –  Gottfried Helms Oct 15 '12 at 14:05
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There are a lot of integers in David Well's The Penguin Dictionary of Curious and Interesting Numbers.

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There's a lot on Wikipedia too (see e.g. en.wikipedia.org/wiki/1000_(number) for 4-digit numbers). –  Douglas S. Stones Oct 17 '12 at 11:45
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One fundamental number in geometry is $2$, the ratio between the diameter and the radius of a circle. And therefore also the ratio between the numbers $\tau$ and $\pi$ (or I should say $\tau$ and $\tau/2$). It is also the ratio of the square on the diagonal of a square and the square itslef. It also appears in various other contexts, too much to enumerate here.

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To prevent a misreading of the notation $ \pi $ is not $ \tau \tau $, but actually $ \tau + \tau $ ... –  Gottfried Helms Oct 15 '12 at 16:34
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The number 6 has many interesting properties. The book Lure of the Integers by Joe Roberts lists among them the following:

  • It is the largest integer which is neither a prime nor the sum of two or more distinct primes
  • It is one of only two integers (and the only composite integer) for which $\phi(n) < \sqrt{n}$ (where $\phi$ is Euler's totient function).
  • Consider sequences defined by bilinear transformations $x_{n+1} = \frac{ax_n + b}{cx_n + d}$. Given $a,b,c,d,x_0$ such that $\forall n \in \mathbb{N}: x_n \in \mathbb{Z}$, the sequence $x_i$ is periodic with period at most 6.
  • Lennes polyhedra of $n$ vertices exist iff $n \ge 6$.
  • It is the smallest perfect number.
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I don't know where I've found this: Proof that all integers are boring: > > 1) Being the greatest of something is exciting. > 2) Being the least of something is boring. > 3) Consider the set of non-boring integers. > 4) In this set there must be least member. > 5) But by 2, this integer is boring. > 6) The integer can not be both boring and non-boring. > 8) Therefore, a least member can not exist. > 9) Since there is no least member, the set of non-boring integers must be empty. > 10) Therefore, all integers are boring. > > <|;-D > > Carl G. $\\$ (Should this be community-wiki ;-) ? –  Gottfried Helms Oct 16 '12 at 11:20
    
@GottfriedHelms I've heard a different version that I think is better (since I would dispute that "being the least of something" is boring). We can show that all natural numbers are interesting. Suppose instead that there is some natural number that is not interesting. There must be a least such number, which is certainly an interesting property, a contradiction. Therefore, all natural numbers are interesting. –  Richard Sullivan Oct 17 '12 at 11:08
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I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." -- G. H. Hardy

This is also the only 4-digit number listed on Wikipedia's "Notable integers".

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