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.
integer constants.
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1"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. – 2012-10-15
5 Answers
Integer constants: what do you want?
- The binomial-coeffcients?
- The Stirling numbers?
- The Eulerian numbers?
- The Bell numbers?
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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0hmm, you asked for "more transcendental examples"... ;-) – 2012-10-15
There are a lot of integers in David Well's The Penguin Dictionary of Curious and Interesting Numbers.
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0There's a lot on Wikipedia too (see e.g. http://en.wikipedia.org/wiki/1000_(number) for 4-digit numbers). – 2012-10-17
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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0To prevent a misreading of the notation $ \pi $ is not $ \tau \tau $, but actually $ \tau + \tau $ ... – 2012-10-15
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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0@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. – 2012-10-17
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".