Questions tagged [diophantine-approximation]

Question 1

Suppose we define a binary sequence $\varepsilon _{k}$ such that it satisfies the equation $$\varepsilon _{k} =\left\lceil \left(\frac{3}{2}\right)^{k}\left( 8-\frac{1}{3}\sum _{j=0}^{\infty }\left(\...

Question 2

Let $z_1 , \dotsc , z_k \in \mathbb{C}$. Is it true that there exists $n \in \mathbb{Z}^+$ such that $\operatorname{Re} ( z_j^n ) \geq 0$ for each $j$? Let $A = [ 0 , 1/4 ] \cup [ 3/4 , 1 ]$. By ...

Question 3

Is it true that the sequence $\{n \sin(n^n/2)\}$ is unbounded but does not tend to infinity? The question was from an introductory calculus class. Here, in this class, “a sequence $\{a_n\}$ tends to ...

Question 4

Introduction : Put $r_n=2^n \{ 1.5^n \} $, i.e : the remainder of $3^n \pmod {2^n}$. If the following inequality holds: $\forall n>1 , r_n<2^n-1.5^n$ (an open problem). It has been proved that ...

Question 5

The Problem : $\forall x,d>0 \quad d\mid 5 \cdot 2^x +9 \implies v_2(d+27) \le x+3 \quad(\star)$ However, I know how to prove this but I'm asking for a new method to prove it ! Why a new method ? ...

Question 6

Consider the equilateral triangular (Eisenstein) lattice $L = \mathbb{Z}\langle a, b \rangle \subset \mathbb{R}^2$ with $a = (1, 0)$ and $b = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. A wrap is ...

Question 7

I'm looking for a known or clean algorithm for the following simultaneous Diophantine approximation problem. I have $n$ real numbers $\vec{a} = (a_1, a_2, \ldots, a_n) \in \mathbb{R}^n$ such that $0 \...

Question 8

I have no idea how to prove if the following series converge: $$ \sum_{n=1}^{+\infty}\frac{(n\varphi-\lfloor n\varphi\rfloor)^n}{n} $$ where $\varphi$ is the golden ratio. None criterion clearly ...

Question 9

Motivation : In a collatz orbit (of odd numbers) , one of the most reasonable questions to ask are : Can we keep dividing consecutively by $2^p$ forever ? ($p>1$) If not, what's the exact number ...

Question 10

Background: The Collatz conjecture is a famous unsolved problem in mathematics. It asserts that, starting from any positive integer, repeatedly applying the rules “if even, divide by 2; if odd, ...

Question 11

I was reading about the Dirichlet and Kronecker approximation theorem. The theorem by Dirichlet says that for any $\theta\in\mathbb R$, $N\in\mathbb N$ there exists some coprime $(h,k)$ where $0<k\...

Question 12

I know that ratios of positive integers can approximate nonnegative real numbers arbitrarily well because for all real numbers $r \ge 0$ and $\epsilon > 0$ there are some integers $a, b > 0$ ...

Question 13

The conjecture : $\forall n>1, d\mid 3^n-2^n \Rightarrow v_2(d+1)<n$ where $v_2(x)$ denotes the $2$-adic valuation of $x$, i.e the highest power of $2$ that divides $x$ e.g : $v_2(12)=2$ since ...

Question 14

We know the basic fact that if $n \equiv 0 \pmod p$ then $|n| \geq p$ (provided $n \neq 0$). Let $\alpha$ be a non-zero algebraic number and suppose that there is a prime ideal $\mathcal{P}$ in $\...

Question 15

I found that $\frac{\pi}{\log(13)}$ is approximated by the fraction $\frac{52378158}{42764081}$ with remarkable precision. Using Wolfram Alpha: $$ \operatorname{N}\left[\frac{\pi}{\frac{52378158}{...

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Questions tagged [diophantine-approximation]

Can we determine whether the binary sequence we obtain through this equation has ever more ones than twice the zeros?

Orbits hitting a specific subset of a multidimensional torus

Is it true that the sequence $\{n \sin(n^n/2)\}$ is unbounded but doesn't tend to infinity?

Are $1,2,5,10$ the only solutions of $3^{n+1} \pmod {2^n} > 2^n-1.5^n$?

A new method of proving : $d \mid 5 \cdot 2^x +9 \implies v_2(d+27) \le x+3$?

Almost-prime lap counts in primitive hexagonal wraps

Seeking fast algorithms for a simultaneous diophantine approximation problem

Convergence of a series involving Fibonacci numbers

A good estimate of $S_k$?

Explicit Baker Constants for Collatz Cycle Constraints?

Approximation of algebraic numbers by quadratic equations

Can ratios of prime numbers approximate real numbers? [duplicate]

Conjecture : $\forall n>1, d\mid 3^n-2^n \Rightarrow v_2(d+1)<n$

Weil height of algebraic number

Why is $\frac{\pi}{\log 13}$ so well approximated by $\frac{52378158}{42764081}$?

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