Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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Exercise 1.12 from Ed Burger's book The Number Jungle.

An earlier exercise asks for a proof of the following result: Corollary 1.9 Let $\alpha$ be a real number and $N$ a positive integer. Then there exists a rational number $p/q$ such that $1\le q\le N$ and $$|\alpha - {\frac{p}{q}}|\le…
student
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Diophantine Approximation on Quadratic Polynomials

Given an integer $a$ which is not a perfect square, I'd like to ask how to perform Diophantine Approximation of $\frac{x^2}{y^2}$ to $a$ where $x$ and $y$ are integers. Specifically, integers satisfying $|\frac{x^2}{y^2}-a|<\epsilon$ are preferred.
Hang Wu
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Equidistribution of $\{p_n^2 \alpha \}$

Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha \}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is equidistributed? I decided to crosspost to…
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Does $lcm\{1,2,...,n\} = \prod_{p\leq n, p\in\mathbb{P}}p^{\lceil \frac{log(n)}{log(p)}\rceil}$?

I am trying to understand Apery's proof of the irrationality of $\zeta(3)$ from start to end, with this document. I apologise for having 2 questions in one, but both are relatively simple (I just need to be sure I completely understand each part of…
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Finding a minima for a linear form with integer coefficients

Some context. This question is aiming to fill gaps in a larger proof, so in a way, it is kind of related to this two other questions (this one and that one) that I asked earlier. But since the problem is formulated in a different way, maybe it will…
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Ordered triplet query

$$x^2 + y^2 + z^2 = 3xyz$$ How many ordered triples $(x,y,z)$ are there that satisfy the above equation. are the only solutions $x=y=z=0$ and $1$? Are there non trivial solutions? I saw this problem in a friends textbook but cannot remember the name…
fosho
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Does Faltings's theorem imply the set of Diophantine equations are decidable?

Actually, we know all Diophantine equations are not decidable. Does Faltings's theorem imply the sets of Diophantine equations are decidable? That is , there is an algorithm that decide whether those equations are decidable. "Let $C$ be a…
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Integrals for semiconvergent approximations to $\pi$

A sequence of semiconvergent approximations to $\pi$ is given by fractions $$3, 4, \frac{7}{2}, \frac{10}{3}, \frac{13}{4}, \frac{16}{5}, \frac{19}{6}, \frac{22}{7}, \frac{25}{8}, \frac{47}{15}, \frac{69}{22}, \frac{91}{29}, \frac{113}{36},…
Jaume Oliver Lafont
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Special Diophantine Equation, Special Pell's Equation

I'm dealing with special kind of "Diophantine Equation". I want to generate all the solutions for $x^2 - aby^2 = a^2 - ab$ given $a,b \in \mathbb{N} $, $a > b$, $a$ and $b$ are co-primes I can easily generate a solution for this problem by putting…
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Mahler's $3/2$ problem

Mahler's $3/2$ problem described here. Am I right in thinking the following would prove it? Show that for every integer $m$, there is some integer $n$ for which the fractional part…
samerivertwice
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How to prove that the distance of $n\sqrt{3}$ to an integer is larger than $\frac{1}{3n\sqrt{3}}$?

How to prove $\forall n \in \mathbb{N}^*$, the following inequation is correct. $$ \min\{n\sqrt{3}-\lfloor n\sqrt{3}\rfloor, \lfloor n\sqrt{3}\rfloor+1-n\sqrt{3}\} \geqslant \dfrac{1}{3n\sqrt{3}} $$
TimeCoder
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Finite amount of pairs (p,q) problem with $ \mid\sqrt{2}-\frac{p}{q}\mid\leq\frac{1}{q^{3}} $

I have no clue how to solve this one. I tried everything. Im helpfull for any tipps. Show that their is a finite amount of Pairs (p, q) ∈ $\mathbb{Z}\times \mathbb{N}$ ,so that : $ \mid\sqrt{2}-\frac{p}{q}\mid\leq\frac{1}{q^{3}} $
Ajax Edm
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Approximation of integer by multiple of irrational number

Obviously, for any $\epsilon >0$, there exist $m,n\in \mathbb{N}$ such that$$|\sqrt{2}-\frac{n}{m}|<\epsilon \; \textrm{.}$$ Is it also true that for all $\epsilon >0$, there exist $m,n\in \mathbb{N}$ such that$$|\sqrt{2}m-n|<\epsilon \;…
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Find the general solutions of $x^2+qy^2-z^2=4n$

I have this diophantine equation and i need the general solution form for it the equation $$x^2+qy^2-z^2=4n$$ some conditions $y\le 0,x\ge 0,z\ge 0$ x,z are even or odd together $n=5y+x-3xy$
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sequence of diophantine approximants of $\pi$

I define the sequence of optimal diophantine approximants of $\pi$ to be the sequence $u_m = \frac{n}{m}$ where $n$ is given by $\min_{\forall n \in \mathbb{N}} |\frac{n}{m}-\pi|$ and we define $\epsilon(m):=\min_{\forall n \in \mathbb{N}}…
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