Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

432 questions
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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 $F_6= \{0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,1\}$. The consecutive differences of…
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Convergence of $\sum_{n=1}^\infty\frac{\cot \varphi\pi n}{n^s}$

Question: $s\in\mathbb C$, Is $$\sum_{n=1}^\infty\frac{\cot \varphi\pi n}{n^s}$$ absolutely convergent, conditionally convergent or divergent, where $\varphi=\frac{1+\sqrt5}2$? TL;DR, my progress It is absolutely convergent if $\Re s>2$ and is…
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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 factors of $2$ on each side? Be patient.) are…
Michael Hardy
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355/113 and small odd cubes

An important approximation to $\pi$ is given by the convergent $\frac{355}{113}$. The numerator and the denominator of this fraction are at the same distance of small consecutive odd cubes. $$\frac{355}{113} = \frac{7^3+12}{5^3-12}$$ Is this a…
Jaume Oliver Lafont
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If $m$ and $n$ are integers, show that $\left|\sqrt{3}-\frac{m}{n}\right| \ge \frac{1}{5n^{2}}$

If $m$ and $n$ are integers, show that $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr| \ge \dfrac{1}{5n^{2}}$. Since $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr|$ is equivalent to $\biggl|\dfrac{ \sqrt{3}n-m}{n}\biggr|$ So I performed the following operation…
K.M
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An Engineer sets out to Prove Fermat's Last Theorem ...

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. But the "proof" that I posted raised some…
10
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M and n are positive integers such that $2^n - 3^m > 0$. Prove (or disprove) that $2^n - 3^m \geqslant 2^{n-m}-1$.

Given that $2^n - 3^m > 0$, I know that $n > m\log_{2}3$ (*). If $2^n - 3^m \geqslant 2^{n-m}-1$, $n>= m + \log_{2}\frac{3^m-1}{2^m-1}$ (**). This is the result when I graph it out ($m$ -> $x$, $n$ -> $y$): https://i.stack.imgur.com/yRCu7.png (*)…
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Asymptotic quality of rational approximations to $\pi$

Dalzell's integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ is case $n=2$ of the generalization $$\int_0^1 \frac{x^{n+2}(1-x)^{2n}}{2^{n-2}(1+x^2)}dx = \frac{p_n}{q_n}-\pi$$ Such an integral gives rational approximations to $\pi$…
10
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Line through the origin mod $1$ visits every sub-cube in $\mathbb{R}^n$.

Let $v$ be the vector $[a_1,a_2,\ldots,a_n]$ where the $a_i$'s are positive integers with $\gcd(a_1,\ldots, a_n)=1$. Let $C$ be the hypercube $[0,1]\times [0,1]\times \cdots \times [0,1]$ in $\mathbb{R}^n.$ Imagine the line given by $vt$ as $t$…
B. Goddard
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Effective equidistribution

I have an irrational number $\alpha$ (in this case, $\alpha=1/(2\pi)$, but hopefully answers will be more general) and I am interested in finding bounds on the size of $$ T=\{k: k\alpha-\lfloor k\alpha\rfloor \in I\} $$ for some interval…
Charles
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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 nonnegative integers) trigonometric polynomial¹…
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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 $\pi \approx \frac{7920}{2521}$. And this is…
9
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Does every $4$-dimensional lattice have a minimal system that's also a lattice basis?

An full $n$-dimensional lattice $\Lambda$ is a discrete subgroup of $\mathbb{R}^n$ (equipped with some norm $\lVert \cdot \rVert$) containing $n$ linearly independent points. If $\Lambda = \{ A z, z\in \mathbb{Z}^n\}$ for $A \in GL(n,\mathbb{R})$,…
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$0<|\sqrt a-\sqrt[3]b|<\epsilon$ for $a,b\in\Bbb Z_+$

I'm trying to solve the following problem: Given $\epsilon>0$, are there positive integers $a,b$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$ ? My solution: given $n\in\Bbb…
9
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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 weaker result $\lim_{n \to \infty} p(n^2+1) = \infty$ can…
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