For questions about approximating real numbers by rational numbers.

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5
votes
2answers
140 views
+50

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 \\ ...
1
vote
1answer
30 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 ...
3
votes
1answer
52 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 ...
2
votes
3answers
53 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
votes
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.
0
votes
0answers
17 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 ...
0
votes
1answer
28 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 $$ |q\alpha - p| < \frac{1}{N}. $$ ...
9
votes
3answers
584 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 ...
4
votes
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 $~p~$ ...
1
vote
0answers
91 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 ...
11
votes
2answers
352 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 $\mathbb{Q[\sqrt{5}]}$. ...
29
votes
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
votes
3answers
183 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
votes
1answer
188 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 ...
3
votes
0answers
132 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
votes
0answers
108 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
votes
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
votes
1answer
474 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 ...
1
vote
0answers
84 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 ...
0
votes
1answer
29 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, ...
1
vote
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 $a_{n+1}:=a_n+a_{n-1}$ ...
1
vote
1answer
240 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
votes
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 ...
-3
votes
2answers
149 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 ...
4
votes
1answer
72 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 ...
75
votes
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 ...
1
vote
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 $G_q$ ...
1
vote
0answers
36 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 ...
32
votes
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 + ...
3
votes
1answer
48 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 $\mathrm{Re}(g)\in\mathbb{Q}$ ...
4
votes
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 ...
8
votes
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
votes
2answers
88 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 ...
2
votes
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
37 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
77 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
59 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 ...
4
votes
1answer
55 views

An uncountable and closed subset of the Liouville Numbers

I am trying to "find" a closed and uncountable subset of the Liouville's numbers. $x\in L$ means that for all $n\in \mathbb{N}$ exists $p,q\in \mathbb{Z}$ with $q>1$ such that $$0<\vert ...
4
votes
0answers
27 views

Approximation of a rational number with values of polynomial

Given two positive integers $p,\space q$, prove that there exist a polynomial $P(x)\in\mathbb{Z}[x]$ and interval $I\subseteq\mathbb{R}$ of length $\frac{1}{q}$ such that $\forall x\in ...
1
vote
5answers
88 views

$10^9 \times \sqrt{3}$ what are first two digits after the decimal point?

Because of floating point error, my computer basically says $10^9 \sqrt{3} \approx 1.73205 \times 10^9$ so that if we ignore the numbers before the decimal point, the fractional part is: $$\{ 10^9 ...
1
vote
1answer
40 views

Best lower bounds on difference between powers

Let $a,b\in\Bbb N$ be fixed. What are the best known lower bounds on $|a^n-b^m|$ for $n,m\in\Bbb N$, provided the difference is not $0$? If $\frac{\log a}{\log b}$ is rational, then $a,b$ are of the ...
2
votes
1answer
48 views

Equation $x^2=y^p+1$

can you help me please for solving this dophantine equation $$x^2=y^p+1$$ and if you can give me a general method to studying such equation $$x²=y^p+t$$ Thanks
0
votes
1answer
45 views

How to find cases where $m^2$ is near to $2^A$?

In another problem here in MSE I ran into the question how I can (practically, in a program) find (positive) integer $m$ such that they are "near" to perfect powers of $2$, so $$ (0 \lt ) \qquad d_m ...
1
vote
2answers
70 views

Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.

Show that $$\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$. I tried many different ways to expand left side and estimate it but always got ...
3
votes
1answer
45 views

Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
0
votes
0answers
43 views

Prove that $p_nq_{n-1}$ - $p_{n-1}q_n=(-1)^{n-1}$ for $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$

Let $p_n$/$q_n$ for $n=0,1,2,..$ be the convergents of $a∈ R$ $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$ $p_n= a_np_{n-1}+p_{n-2}$ $q_n= a_nq_{n-1}+q_{n-2}$ I need to prove that $p_nq_{n-1} - ...
13
votes
3answers
881 views

Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
21
votes
1answer
519 views

Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
5
votes
1answer
117 views

Intuition behind proof in the Rudin book that there is no largest/smallest real number. [duplicate]

In Rudin's Principles of Mathematical Analysis (3rd ed), he proves (at the very beginning: example 1.1) that the set $A$ of all positive rationals $p$ such that $p^2<2$ contains no largest number ...
1
vote
1answer
22 views

Diophantine approximation with integer vectors

I would like to determine whether or not there exists ${\beta > 0}$ and ${\gamma \geq 2 }$ such that ${ \forall (m_{1},m_{2}) \in \mathbb{Z}^{2} \setminus (0,0) }$, one has the inequality $$ ...