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34
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
23
votes
2answers
825 views

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
21
votes
3answers
2k 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.
15
votes
1answer
201 views

Does $ \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit?

The entire question is essentially given in the title: For $ n $ a positive integer, does the sequence $ x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit as $ n \to \infty $? ...
15
votes
1answer
296 views

Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?

Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
14
votes
1answer
319 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
13
votes
1answer
520 views

Is there a 'far' irrational number?

I recently learned of so-called 'far' numbers at a talk. In the talk, it was proven that there is a dense subset of the interval $[0,1]$ of far numbers (however, far numbers were only a minor point of ...
13
votes
5answers
3k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
12
votes
3answers
762 views

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

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
12
votes
3answers
519 views

When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer ...
11
votes
0answers
306 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that ...
10
votes
1answer
377 views

For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it. I have an always-nonnegative (on the ...
10
votes
0answers
210 views

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

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 ...
9
votes
1answer
171 views

What is the sum of the squares of the differences of consecutive element of a Farey Sequence

A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example $F_6= ...
8
votes
2answers
168 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
8
votes
0answers
265 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 ...
8
votes
1answer
95 views

Do best lower approximations of a quadratic irrational always form a linear recurrence sequence?

Let $\theta$ be an irrational number and let $$ {\cal L}= \bigg\lbrace (a,b) \in {\mathbb Z} \times {\mathbb N}^{*} \bigg| \frac{a}{b} \leq \theta \bigg\rbrace $$ and $$ {\cal B}= \bigg\lbrace ...
7
votes
5answers
704 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
7
votes
1answer
160 views

Diophantine equation $x^3+x+y^3+y = z^3 + z$ for $x,y,z>0$

Consider the Diophantine equation $x^3+x+y^3+y = z^3 + z$ for positive integer $x,y,z$. I tried small values and got some near equalities : $(5,6,7)$ and $(12,16,18)$ are true up to value $2$. $( 5^3 ...
6
votes
2answers
552 views

Applying the Thue-Siegel Theorem

Let $p(n)$ be the greatest prime divisor of $n$. Chowla proved here that $p(n^2+1) > C \ln \ln n $ for some $C$ and all $n > 1$. At the beginning of the paper, he mentions briefly that the ...
6
votes
1answer
252 views

Cube roots don't sum up to integer

My question looks quite obvious, but I'm looking for a strict proof for this. (At least, I assume it's true what I claim.) Why can't the sum of two cube roots of positive non-perfect cubes be an ...
6
votes
1answer
153 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
5
votes
2answers
151 views

The real part of $z^n$

Prove that $${\displaystyle\lim \limits_{n \to +\infty}{|r^ncos(nθ)|}}=+\infty,$$ where $n$ is integer, $r>1$, $θ/π$ is irrational. I got this problem from here $1+x+\ldots+x^n$ perfect square , ...
5
votes
1answer
172 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...
5
votes
1answer
132 views

badly approximated numbers on the real line form a meagre set

Let $S$ be the set of real numbers $x$ such that there exist infinitely many (reduced) rational numbers $p/q$ such that $$\left\vert x-\frac{p}{q}\right\vert <\frac{1}{q^8}.$$ I would like to prove ...
5
votes
0answers
63 views

finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
4
votes
1answer
248 views

Discrete rational rotations on the two dimensional torus

It is well known (Kronecker's Theorem) that "irrational rotations" are dense on $[0,1)$. That is, the set $$ \{ x+nr\mod 1 : n \in \mathbb{N} \} $$ is dense on $[0,1)$, provided that $r$ is ...
3
votes
1answer
46 views

Approximating the square root of two with fractions

I would like to prove that there exist only finitely many $m, n \in \mathbb{N}$ satisfying $$\left | \sqrt{2} - \frac{m}{n} \right | < \frac{1}{4n^2}.$$ Any thoughts? Thank you for your help.
3
votes
1answer
54 views

Minkowski's theorem for non-symmetric convex bodies

Minkowski's theorem for convex bodies states that every convex, symmetric subset of $\mathbb{R}^d$ whose volume is larger than $2^d$ contains a non-zero integer point. All the proofs I've seen rely on ...
3
votes
2answers
71 views

Where can i find resourses to study this algebraic number theory?

Where can i find material to study depper the farey fractions (continued fractions)? I triying to solve problems like these: 1.- Show that two consecutive convergent at least one of them satisfy: ...
3
votes
1answer
184 views

Kronecker's theorem - converse

I don't know how to prove that Kronecker's theorem is false if $\alpha_{1}$, $\alpha_{2}$, $\dots$, $\alpha_{M}$ are not independent. Kronecker's theorem Suppose that $\alpha_{1}$, ...
3
votes
1answer
73 views

diophantine approximation

For which $\alpha \in \mathbb R$ can one say that $\forall \epsilon > 0$ there $\exists N \geq 1$ such that $\forall i \in \mathbb N$ one has that some $n\in \{1, \dots, N\}$ is a solution to $$ ...
3
votes
2answers
191 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are (1,3) (11, 4), (19, 5), (29, 6), (61, 8) and other one is I don't know. This ...
3
votes
2answers
75 views

Bound to the Number of Solutions of the Thue-Siegel-Roth Theorem

I have to prove for an assignment that if the partial quotients of a number $x$ are $\frac{p_n}{q_n}$, and $$\limsup\frac{\log\log (q_n)\sqrt{\log n}}{n}=\infty$$ then $x$ is transcendental. I do not ...
3
votes
0answers
131 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
2
votes
3answers
171 views

Sum of roots is integer

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
2
votes
2answers
180 views

An experiment was performed n times. The exact success rate rounds to 85.8%. What is the minimum value of n?

An experiment was performed n times. The success rate was given as 85.8% Clearly, if there were 858 successes out of 1000 trials it would give that percentage. However, the percentages are rounded to ...
2
votes
2answers
121 views

How find the value $\beta$ such $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Find all positive real number $\beta$,there are infinitely many relatively prime integers $(p,q)$ such that $$\left|\dfrac{p}{q}-\sqrt{2}\right|<\dfrac{\beta}{q^2}$$ maybe this problem background ...
2
votes
2answers
233 views

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is "uniform" in the sense that all finite digit strings occur with the "statistically expected" frequency. I read a sentence somewhere ...
2
votes
2answers
165 views

Prove that for any give sequence of digits, there is a perfect square starting with that sequence

Prove that for any give sequence of digits, there is a perfect square starting with that sequence. With more details, prove that for $\forall a\in \mathbb{N}$, such that ...
2
votes
1answer
43 views

Simultaneous Diophantine approximation

Is the following statement true? For all $\alpha,\beta\in \mathbb R$ and for all $\varepsilon \in \mathbb R_{>0}$, there exist $a,b,c\in \mathbb Z$ such that $|a-c\alpha|<\varepsilon$ and ...
2
votes
2answers
485 views

Is $\sin^2 n$ bounded away from zero?

Can one find a number $m$ such that $\sin^2 n \geq m > 0 $ for all integers $n$? By continuity of $\sin x$, it is enough to say that $|n - k \pi| \geq m^{\prime} > 0$ for all integers $n,k$. ...
2
votes
2answers
90 views

Criteria for irrationality of real numbers

Concerning the criteria for irrationality: Theorem says "A number $\alpha$ is irrational if and only if for every $\epsilon >0$ there exist integers $h$ and $q$ such that $0 < | q\alpha - h | ...
2
votes
1answer
91 views

Approximate solution of a diophantine equation

Consider the Diophantine equation $P(x)=y^2$, where $P$ is a (nonconstant) polynomial with integer coefficients and $x$ and $y$ must be integers. For $\varepsilon \gt 0$, I say that an integer $x$ is ...
2
votes
1answer
164 views

Show that a rational number has no good rational approximations

This is homework question. The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$. What we need to ...
2
votes
1answer
117 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 ...
2
votes
1answer
263 views

Dirichlet approximation theorem and BA numbers

I think some version of the following definition given in "Diophantine Approximations" by W.M. Schmidt: Let $\sigma \in (0,1)$ and $x \in \mathbb{R}$. Then $x$ is $\sigma$-improvement for ...
2
votes
1answer
44 views

Diophantine approximation problem

Denote $||x||=Min(x-[x],1-(x-[x]))$,it means the minimum distance between x and an integer. Can we find a fast algorithm to get a natural number $n$ that satisfies $$||na||<p,||nb||<p,...$$ ...
2
votes
0answers
40 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
votes
0answers
33 views

A generalisation of Roth's result on Diophantine approximation?

It is a celebrated result of Roth that algebraic numbers cannot be approximated by rationals too accurately: $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert_{\mathbb{R}/\mathbb{T}}}$ Theorem ...