integer constants.

Question

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.

Answer 1

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?

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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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Answer 2

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.

Answer 3

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.

Answer 4

There are a lot of integers in David Well's The Penguin Dictionary of Curious and Interesting Numbers.

Answer 5

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".