All Questions
Tagged with ds.dynamical-systems nt.number-theory
140 questions
6
votes
0
answers
448
views
Are there always at least *five* divisions?
@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n \;\text{...
- 1,375
31
votes
4
answers
2k
views
A Collatz-like function that bifurcates on primes
This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...
- 151k
22
votes
4
answers
2k
views
A question on Collatz's conjecture:proportion of "low flying" orbits
Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k (...
- 6,349
2
votes
0
answers
248
views
Linear forms with best approximation vectors lying in a subspace
Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in \mathbb{R}^...
- 616
4
votes
2
answers
602
views
Rate of convergence of an irrational rotation
Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$.
If we assume that $\alpha$ is irrational, then there exists an increasing ...
- 43
34
votes
2
answers
2k
views
Does iterating a certain function related to the sums of divisors eventually always result in a prime value?
Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...
- 337
1
vote
0
answers
122
views
square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces
I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant:
...
- 22.8k
21
votes
4
answers
2k
views
Prime factorization "demoted" leads to function whose fixed points are primes
Let $n$ be a natural number whose prime factorization is
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$
Define a function $g(n)$ as follows
$$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \;,$$
i.e., exponentiation is "...
- 151k
11
votes
1
answer
462
views
A strengthening of base 2 Fermat pseudoprime
If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$,
$k$ divides ${n-1 \choose 2k-1}$ because of the identity
${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether
an ...
- 487
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
18
votes
3
answers
1k
views
Not-lonely runners
The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...
- 151k
23
votes
0
answers
896
views
Base change for $\sqrt{2}.$
This is a direct follow-up to Conjecture on irrational algebraic numbers.
Take the decimal expansion for $\sqrt{2},$ but now think of it as the base $11$ expansion of some number $\theta_{11}.$ Is ...
- 96.4k
30
votes
2
answers
2k
views
Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$
Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ \...
- 19.6k
2
votes
0
answers
118
views
random maass waveforms
Let $H$ be the upper half complex plane and $\Gamma$ a discrete subgroup of $SL_2(\mathbb{Z})$ such that the volume of of $\Gamma \backslash H$ is finite. There is a conjecture of Berry that Maass ...
- 101
7
votes
1
answer
4k
views
Beyond Collatz: A $5n+1$ conjecture? [closed]
Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
- 254
11
votes
3
answers
896
views
Integer dynamics hitting infinitely many primes
I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, ...
- 960
18
votes
1
answer
653
views
Can the expansion of a large integer in all bases consist of almost all zeroes?
Let $n$ be a positive integer. Given an integer base $b\ge 2$, let $C_b(n)$ be the number of non-zero digits in the expansion of $N$ in base $b$. Further, let $M(n)=\max\{C_b(n):b\ge 2\}$ be the ...
- 4,660
4
votes
2
answers
484
views
Algebraicity of the "outer" boundary of the Mandelbrot set
Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as
$$
t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu \...
- 3,897
8
votes
1
answer
874
views
"Explicit" examples of Irrational numbers very well approximated by rational numbers
This question relates to this one and that one.
Some background
In the setting of discrete holomorphic dynamics (say, Julia sets)
an irrational $\lambda$ is said to be well approximated by rational
...
- 5,428
30
votes
3
answers
8k
views
Status of the 196 conjecture?
A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not ...
- 1,363
3
votes
0
answers
309
views
A Dedekind Eta trajectory / horocyclic flow (Reference request)
I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\...
- 10.5k
23
votes
1
answer
4k
views
The Dedekind eta function in physics
This interesting little fellow (a nice introduction is the video "Mock Modular Forms are Everywhere" by Cheng and Felder) popped up in some operator algebra (Witt / Virasoro Lie algebra) I ...
3
votes
1
answer
445
views
Bohr sets, Coin-flip sets and Roth's theorem
I have been learning about Roth's theorem, trying to understand how Fourier series and dynamical systems (or even graph theory and binary sequences)are involved in counting arithmetic sequences in ...
- 22.8k
48
votes
1
answer
2k
views
A function whose fixed points are the primes
If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...
- 3,081
10
votes
3
answers
313
views
The identity element of a compact group is a limit point of any "polynomial sequence"
Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial ...
- 235
9
votes
3
answers
3k
views
Simultaneous diophantine approximation
Let $r(x)$ be the function $x$ mod $1$, i.e. $x$ minus its floor.
Now let $m$ be a given positive integer, and $c$ a vector in $\mathbb{R}^m$ whose components are linearly independent over $\mathbb{Q}...
- 409
5
votes
0
answers
282
views
Limits of $p/\ln p - q /\ln q$, $p, q$ prime
Is there any $\alpha>0$ for which there are known to exist two sequences of primes, $(p_i), (q_i)$ such that
$$\alpha = \lim_{i\to\infty} \left(p_i/\ln p_i - q_i /\ln q_i\right)\ ?$$
The ...
- 17.6k
17
votes
2
answers
886
views
Branches of the Fibonacci Word Tree
The Fibonacci word starts from $0$ subject to the rules $0 \mapsto 1, 1 \mapsto 01$ (or some variant thereof). The come from cutting sequences of the torus of a line of golden ratio slope. It is a ...
- 22.8k
12
votes
5
answers
2k
views
Computing the centers of Apollonian circle packings
The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to ...
- 22.8k
0
votes
1
answer
1k
views
Implication for cycles (of some length $m$) in Collatz-type problems: typical ratio between largest and smallest element?
Background
Consider Collatz-type problems of the form $an + 1$, where $a > 2$ is a positive, odd integer (e.g., $3n + 1$, $5n +1 $, $7n + 1$, etc.). For convenience, automatically divide by two.
...
4
votes
0
answers
658
views
Riemann hypothesis and action principle [closed]
Hello,
I would like to know whether the Riemann hypothesis could be a consequence of some kind of action principle: in other words, can the equation $\zeta(s)=0$ be interpreted as the formulation of ...
- 6,974
7
votes
1
answer
382
views
Are gaps in $n^2\sqrt{2}$ poissonian?
I would like to know gaps about the sequence $n^2 \sqrt{2} \mod 1$. Van der Corput's trick shows that $n^2 \sqrt{2}$ is equidistributed on the circle. For large $N$, the fraction
$$ \frac{\# \{ 1 \...
- 22.8k
66
votes
4
answers
4k
views
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
- 25.1k
4
votes
1
answer
1k
views
Are all Nilmanifolds quotients of Heisenberg Group
I've been reading some wonderful blog entries where Terry Tao and Ben Green prove some generalizations of Weyl Equidstribution using a "higher" Fourier Analysis. Unfortunately, all the information I ...
- 22.8k
7
votes
3
answers
2k
views
How can one express the Dedekind eta function as a sum over the lattice?
The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...
- 1,821
9
votes
3
answers
591
views
subtracting greatest possible prime
Let $A$ be a set of positive integers, $\min A:=a_0$. For $x\geq a_0$ define $f(x)=x-a$, where $a\leq x$, $a\in A$ is the maximal possible. Then for a positive integer $x$ the iterations $x$, $f(x)$, $...
- 109k
13
votes
1
answer
1k
views
Conjectures on iterated polynomial maps on finite fields
Let $p$ be a prime, and consider the sequence $x_0, x_1, \dots$ of elements of the finite field $\mathbf F_p$ given by $x_0 = 0$ and $x_{i+1} = x_i^2 + 1$ for all $i \ge 0$. This sequence must ...
- 233
14
votes
2
answers
1k
views
Polygonal billards programs
I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
(source)
It was a good exercise, but at this point I ...
- 22.8k
37
votes
1
answer
1k
views
A question of Erdős on equidistribution
In his book Metric Number Theory, Glyn Harman mentions the following problem he attributes to Erdős:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
$$\lim_{N\...
- 2,151
4
votes
2
answers
648
views
Gaps in nx (mod 1)
It is known that if you choose n point at random on S1 = [0,1], the nearest neighbor spacings between the points are exponentially distributed with mean 1.
For example, two of our n points could be ...
- 22.8k