For questions about approximating real numbers by rational numbers.

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0answers
32 views

Numerators of close rational approximations to $\pi$

Define a rational number $\frac{p}{q}$ to be $\epsilon$-close to $\pi$ if $$\left|\pi - \frac{p}{q}\right| \leq \frac{1}{q^{\mu-\epsilon}}$$ where $\mu$ is the irrationality measure of $\pi$, ...
1
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1answer
112 views

Siegel's Lemma for two solutions

Consider the homogeneous diophantine equation $$ax_1+bx_2+cx_3=0$$ over $\mathbb{Z}$ with a, b, c coprime. (A version of) Siegel's Lemma states, that there exists a non-trivial solution $x$, such ...
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4answers
47 views

Show that $\inf \{ m+n\omega: m+n\omega>0,and,m,n\in{Z}\}= 0$, where $\omega>0 $ is irrational.

Let $\omega\in\mathbb {R}$ be an irrational positive number. Set $$A=\{m+n\omega: m+n\omega>0,and,m,n\in{Z}\}.$$ Show that $\inf{A}=0.$ How should I start this problem? I don't get this problem.
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0answers
42 views

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 $n=5y+x-3xy$
2
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0answers
57 views

positive integer solutions of $6xy+5x+5y=n$

Can any one help me with this? Determine all Positive integer $(x,y)$ such that $2 \le x \le y$ and $6xy+5x+5y=n$ Please do not solve as $(6x+5)(6y+5)=6n+5^2$, I need a more helpful method. we have ...
0
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1answer
65 views

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 $\...
2
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1answer
59 views

Question on the property of algebraic number [closed]

Given $x$ an algebraic number of degree 2. Anyone help me to prove $$ |\sin(n\pi x)| \geq \frac{\mathrm{const.}}{n} $$ for all $n$ ?
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1answer
46 views

$d^{1/3}$ badly approx. $\iff$ $d^{2/3}$ is badly approx.

Let $d$ be an integer, that is not a cube of any integer. Show that $d^{1/3}$ is badly approximable iff $d^{2/3}$ is also. Badly approximable means that, there is a contant $C$ s.t. $\lvert \beta-\...
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0answers
46 views

Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
2
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1answer
57 views

Hardy- Littlewood Circle Method

I'm currently trying to get to grips with the Hardy Littlewood circle method so I'm working through Vaughan's book. In the past I've been very bad for leaving a point behind if I don't follow it so I'...
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1answer
38 views

A step on the proof of Liouville's theorem on approximation

I'm having trouble following one step in the proof of Liouville's theorem on approximation of real algebraic numbers, from Murty and Rath's book "Transcendental Numbers". The step is: $$|\alpha-\...
4
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0answers
38 views

Prove that there are only finitely many rational numbers $p/q$ satisfying $|\frac{p}{q}-\sqrt[d]{b}|\leq \frac{C}{q^3}$.

This is a problem from Silverman & Tate's Rational Points on Elliptic Curves. The following is the Diophantine Approximation theorem by Thue which is proved in Chapter 5: Theorem. Let $b$ be a ...
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0answers
35 views

Solution of diophantine equation with lowest c

Lets say I have a diophantine equation , aX - bY = c Now, for some (a,b,c) I may not have any integer solution at all. But lets say , I write the equation in this way , aX - bY = c + p p is an ...
6
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2answers
204 views

Kernel of a Vandermonde like matrix

I am wondering how to show that the following matrix has trivial kernel: $$\begin{bmatrix} 1&1&1&1&1&1 \\ s_1&s_2&s_3&s_4&s_5&s_6 \\ s_1^2&s_2^2&...
1
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1answer
34 views

Product of denominators exceed $n$ in Farey sequence

Why in the $n$-th Farey sequence the product of the denominators of $2$ adjacent fractions exceed $n$ ($0$ and $1$ are excluded) ? I have a theorem of Hurwitz which states: For every irrational ...
3
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1answer
55 views

Infinitely many rationals with $|a-\frac pq|<\frac1{q^2}$

If $a$ is irrational, there are infinitely many $\frac pq$ s.t $|a-\frac pq|<\frac1{q^2}\tag1$ I have the proof but don't understand it: Take a finite set of rationals $S$ then for sufficiently ...
2
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3answers
63 views

Rational approximation of square roots

I'm trying to find the best way to solve for rational approximations of the square root of a number, given some pretty serious constraints on the operations I can use to calculate it. My criteria for ...
0
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0answers
14 views

Recursive definition of Minkowski ?(x) function

There is a fact that $?(\frac{a+b}{c+d}) = (?(\frac {a}{c}) + ?(\frac{b}{d})) / 2$ if a/c and b/d are adjanced elements of Farey sequence. How to prove it? I don't have any ideas at all.
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0answers
20 views

Intuition in understanding Minkowski question mark ?(x) function

There are 3 definitions of Minkowski function: 1) If $[a_0; a_1, ...]$ is a continuous function representation of n 2) Consider the different ways of interpreting an infinite string of bits ...
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1answer
30 views

Thue-Siegel-Roth Type Theorem

Dirichlet's approximation theorem says that for every real $\alpha$ and every positive integer $N$, there exist integers $p,q$ with $1 \leq q \leq N$ such that $$ |q\alpha - p| < \frac{1}{N}. $$ ...
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3answers
602 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
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1answer
38 views

Finding irrational numbers in given interval

If $~\xi~$ is irrational number then it is known that the set $~\{ p \xi + q ~ | ~ p,q \in \mathbb{Z} \}~$ is dense in $~\mathbb{R}$. Thus given some reals $~a~$ and $~b~$ one can find integers $~p~$ ...
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0answers
96 views

Why does $63725\pi$ give four approximations to $\pi$?

The fraction $\frac{355}{113}$ was first used for approximating $\pi$ by the Chinese mathematician and astronomer Zu Chongzhi in the 5th century (see Milü). An isolated case? The almost-integer $113\...
11
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2answers
356 views

How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...
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4answers
4k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
10
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3answers
189 views

Why does $29^2 : 31^2 : 41^2$ have a close integer approximation with small numbers?

"Everybody knows" that such coincidences as $$2\times2\times\overbrace{41\times41} = 6724 \approx 6728 = 2\times2\times2\times\overbrace{29\times29}$$ (And why did I bother with the first two ...
2
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1answer
190 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
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0answers
136 views

A particular Diophantine approximation of $\pi/2$.

Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$ After some computations, I was led to the following question: let $p_n,q_n$ be two sequences of natural ...
6
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0answers
109 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
0
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1answer
32 views

Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
14
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1answer
488 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
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0answers
85 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
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1answer
30 views

Show that $\{p/q:|\frac{\sqrt5+1}{2}-p/q|<\frac{1}{\sqrt{5}q^2}\}=\{\frac{F_{2n+1}}{F_{2n}}:n\in\mathbb N\}$

Show that $\{p/q:|\frac{\sqrt5+1}{2}-p/q|<\frac{1}{\sqrt{5}q^2}\}=\{\frac{F_{2n+1}}{F_{2n}}:n\in\mathbb N\}$, where $F_n$ is the $n$-th Fibonacci number $2$ things to show $1$st, $\frac{F_{2n+1}}...
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2answers
28 views

General solution expressed in $a_0$ and $a_1$ of a Fibonacci-like sequence?

What is the general solution expressed in $a_0$ and $a_1$ of a Fibonacci-like sequence ? I mean if $a_0,a_1$ are given and $a_{n+1}:=a_n+a_{n-1}$ $(\begin{array}{cc}a_n&a_{n-1}\end{array})=(\...
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1answer
263 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
2
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1answer
114 views

Series and integrals for inequalities and approximations to $\log(n)$

The following series and integrals relate log(2) to its third and fourth convergents, $\frac{2}{3}$ and $\frac{7}{10}$. $$\begin{align} \log\left(2\right)-\frac{2}{3} &= \sum_{k=1}^\infty \frac{...
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2answers
156 views

A series of positive terms to prove $\pi>\frac{333}{106}$

This is a consequence of the answer to that question. A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms $$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty \frac{...
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1answer
73 views

Problem with inequality: $ \left| \sqrt{2}-\frac{p}{q} \right| > \frac{1}{3q^2}$

Prove that for for all $p,q\in \mathbb{Z}$, $q>0$ we have: $$ \left| \sqrt{2}-\frac{p}{q} \right| > \frac{1}{3q^2}. $$ To be honest, I do not know where to start - any help would be appreciated....
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14answers
10k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
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1answer
19 views

Diophantine approximation and covers

Suppose $\alpha > 2$. Let F be the set of real numbers $x \in [0,1]$ for which the inequality $||qx|| \le q^{1-\alpha}$ is satisified by infinitely many positive integers q. For each q, let $G_q$ ...
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0answers
48 views

What does $\log^3$ stand for in this paper by K. Győry?

In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$. Namely, consider a ...
33
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0answers
1k views

Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^...
3
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1answer
49 views

finding rational complex numbers in a disk with least denominators

Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$: How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ ...
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1answer
64 views

Proving that sequence elements satisfy inequality involving $\mod{1}$

I'm trying to prove that $n(3-\sqrt{8}) \; (\!\!\! \mod{1}) < m(3-\sqrt{8}) \; (\!\!\! \mod{1})\;\; \forall \; m < n \;|\;n,m \in \mathbb{N}$ $\leftrightarrow n \in \{a_k:(a_k=6a_{k-1}-a_{...
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1answer
112 views

Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
3
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2answers
93 views

Question on a constructive proof of irrationality of $\sqrt 2$

Here is the constructive proof of $\sqrt 2 \not \in \mathbb Q$ found on this page : Given positive integers $a$ and $b$, because the valuation (i.e., highest power of 2 dividing a number) of $2b^2$...
2
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0answers
35 views

Average of a certain diophantine function

Yesterday I was browsing math.se and came across this question. It was answered by a few people and the best answer was already accepted so I just read the question and the solutions to it. Then I ...
3
votes
0answers
39 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
6
votes
2answers
79 views

Specification of Hurwitz's Theorem

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, the equation $$\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$ has infinitely many solutions $(p,q)$ ...
5
votes
1answer
73 views

Can finite sums of two numbers come arbitrarily close to zero?

Given two real numbers $a$ and $b$, define an $a$-$b$-sum as a finite sum of $a$'s and $b$'s, i.e. a sum: $$m\cdot a + n\cdot b$$ where $m,n$ are non-negative integers. Is there a pair of numbers $a&...