User markus redeker - MathOverflow

Are there results in "Digit Theory"?

2013-03-21T10:46:41Z

Results about numbers that are related to their decimal representation are usually confined to recreational mathematics. There I have seen mainly questions about individual numbers, like finding a number that is the sum of the cubes of its digits. But I have not yet seen a systematic study of such questions.

A result that prompted me to asked this question was the following from Kurt Hensel's "Zahlentheorie". There he shows that the exponent of highest power of a prime $p$ that divides the number $n$ is $$\frac{s_{n-1} - s_n + 1}{p - 1},$$ where $s_n$ is the sum of the digits in the representation of $n$ with base $p$. From this he derives that the highest power of $p$ that divides $n!$ is $$\mu_n = \frac{n - s_n}{p - 1}.$$ This is a different and much simpler formula than the representation $\mu_n = \sum_{k = 0}^\infty \left\lfloor \frac{n}{p^k} \right\rfloor$ by Legendre, which I usually see in the literature.

I would like to know whether there are more results of this type. To be more precise, I define the not-yet-existing field of "Digit Theory" as the study of number-theoretic questions with help of the positional representation of numbers, and the study of the positional representation itself.

So are there already systematic treatments of "Digit Theory" or of parts of it? Are there other interesting results like that one above?

Answer by Markus Redeker for Visualizing a graph

2013-03-13T12:24:29Z

You could try yEd. It can import data in various formats, and there is an image on their web site of a graph with 13,500 nodes and 26,000 edges, which is much larger than your graph.

Answer by Markus Redeker for Should one attack hard problems?

2013-02-27T10:00:24Z

When thinking about attacking a hard problem one should ask: Do I have a tool or an idea that other people who attacked the problem did not have? If so, one should give it a try. If not, one will be most probably not better than others.

Answer by Markus Redeker for What are some examples of "chimeras" in mathematics?

2012-05-28T09:16:56Z

In quantum mechanics, the set of possible energies for a system of two attracting particles, say an electron and a proton, consists of a discrete part (the bound states) and a continuous part (the unbound states).

So this is quite a "natural" example. To make it more "mathematical" one can express it as an eigenvalue problem.

Generalised de Bruijn Graph

2012-05-25T14:41:45Z

I have encountered sets of the following type, consisting of words over a finite aphabet $A$.

If $S$ is such a set, then

$S$ is finite,
No word in $S$ is part of another element of $S$, and
every infinite sequence $\dots a_{-2} a_{-1} a_0 a_1 a_2 \dots$ of elements of $A$ can be covered by elements of $S$ in the following sense: for every $k \in \mathbb Z$ there are numbers $i$, $j$ with $i \leq k \leq j$ such that $a_i a_{i + 1} \dots a_j \in S$.

Now I take the elements of $S$ as vertices of a directed graph $G$ with loops; an edge between two words $w_1$, $w_2 \in S$ exists exactly then when there is a sequence of elements of $A$ in which $w_1$ is followed directly by $w_2$.

If $S = A^n$, the set of all words of length $n$, then $G$ is a de Bruijn graph. But what about the general case? Has anyone written about it, or do you know anything interesting about such graphs?

Notation for ends of a string

2012-04-13T09:34:09Z

I work now a lot with strings of characters and other finite sequences and found that I need many times a good notation for "cutting the end" a string. If $a$ is a finite sequence and $a'$ is its right end, then there is a sequence $x$ such that $a = x a'$ (where the product is defined by concatenation). What I would like to have is a suggestive notation for $x$: $x$ might be $a$ "minus" $a'$, or $a$ "divided by $a'$, if we view the set of all strings as a semigroup. I have therefore thought of writing $x$ as $a / a'$ but am not completely happy with it. Obvoiusly I need also a notation for "cutting at the left end"; by continuing with the previous idea the notation $a' \setminus a$ could stand for the solution of $a = a' x$, but it collides a bit with the notation for set difference.

So far I have thought, and my question is: Is there a good notation for these concepts already in use in some areas of mathematics or computer science, or has someone else already defined a suggestive notation?

P.S. And I also would like to have a notation for the "overlapping concatenation" of strings: If $a = a'x$ and $b = xb'$, what is $a'xb'$?

Comment by Markus Redeker

2013-04-06T10:46:43Z

This is because in complex analysis and differential geometry a point is also a variable (in $\mathbb C$ or $\mathbb R^n$, respectively), and such variables are by convention lowercase.

Comment by Markus Redeker

2013-03-25T09:45:54Z

Thanks for bringing Bunjakovskiĭ to our attention. Do you refer to the first or the second formula? It seems that "digit theory" has existed for a long time in the mathematical underground, without ever becoming really respectable.

Comment by Markus Redeker

2013-03-22T09:57:01Z

This of course requires a simple direct proof for the formula for $\mathrm{ord}_p(n!)$. Do you know one?

Comment by Markus Redeker

2013-03-11T11:11:33Z

Yes, but he started to work at the problem when it became known that there was a connection. At this point he knew that he had an approach that earlier researchers had not.

Comment by Markus Redeker

2013-02-27T13:18:02Z

@Frank: But you can often see that a method is new. Examples are Andrew Wiles' use of elliptic curves for Fermat's problem or Heinrich Heesch's idea of using computers to attack the Four Colour Problem.

Comment by Markus Redeker

2013-01-24T18:38:23Z

@Rasmus: Your notation will become a bit less useful once $a$ and $b$ are replaced by more complex terms. In that case Dijkstra's notation is better.

Comment by Markus Redeker

2013-01-24T18:32:22Z

This can also be done by writing $\int_x^{2\pi} dx\, \sin x$ etc., which is a little bit shorter. I have often seen this in physics.

Comment by Markus Redeker

2013-01-21T11:44:05Z

Hexahedral graph 5: Take a tetrahedron, glue on one of its faces another tetrahedron and on that one a third tetrahedron, such that the three tetrahedra have one edge in common. Visualisation worked for me because I recognised the graph of the tetrahedron in the heahedral graph. This may be part of a technique.

Comment by Markus Redeker

2012-12-22T13:53:50Z

"Condensation of Singularities" was invented by Hermann Hankel. Basically it is the following: You have a function $f$ that has a singularity at 0. Make a periodic function $g$ from it that has this singularity at every integer. Then form the sum $G(x) = \sum_{n=0}^\infty a_n g(n x)$, with the $a_n$ chosen such that the sum converges. Then $G$ has this singluarity at every rational point. This can be done with mny types of singularities, provided that they are well-behaved enough under addition.

Comment by Markus Redeker

2012-12-19T13:59:15Z

What are the conditions on $p$ and $q$ in (4)?

Comment by Markus Redeker

2012-12-10T16:33:05Z

This does not really qualify, because Euler re-proved an older result. But didn't he get more results from his proof than Euclid's theorem?

Comment by Markus Redeker

2012-04-14T08:24:42Z

@Joel: Thanks for the hint - I have added the tag.