For questions about approximating real numbers by rational numbers.

# Questions tagged [diophantine-approximation]

432 questions

**127**

votes

**1**answer

### Motivation of irrationality measure

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 $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$$ So the series for $e^{-1}$ is…

Damien

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**93**

votes

**17**answers

### 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 infamous values, $\large\pi$ and $\large e$
The closest…

Nick

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**63**

votes

**4**answers

### 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 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1…

user55114

**61**

votes

**5**answers

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

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

user11135

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**54**

votes

**1**answer

### 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 $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq…

Jack D'Aurizio

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**45**

votes

**4**answers

### Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound?

Is the sequence $n+\tan(n), n \in\mathbb{N}$ bounded below?
Intuitively I think it is not bounded below, but I have no idea how to prove it. It is like a Diophantine approximation problem, but most theorems seem to be too weak.

MathEric

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**43**

votes

**5**answers

### 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 series The Simpsons.
Thanks to Andrew Wiles, we all…

Stefan4024

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**36**

votes

**5**answers

### How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can
$S(n) = \sum_{k=1}^n \sqrt{k}$
be to an integer?
Is there some $f(n)$ such that,
if $I(x)$ is the closest integer to $x$,
then $|S(n)-I(S(n))|\ge f(n)$
(such as $1/n^2$, $e^{-n}$, ...).
This question was inspired by the recently…

marty cohen

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**33**

votes

**1**answer

### Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer.
There are some known ways to construct such examples in a systematic way. One is through the use of certain algebraic numbers called Pisot numbers. These numbers…

user346361

**32**

votes

**1**answer

### $\pi^4 + \pi^5 \approx e^6$ 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 approximate just about any number. In formal…

cactus314

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**30**

votes

**1**answer

### 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 criteria for irrationality. When applying some of…

Chazz

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**30**

votes

**3**answers

### Is the sequence $(B_n)_{n \in \Bbb{N}}$ unbounded, where $B_n := \sum_{k=1}^n\mathrm{sgn}(\sin(k))$?

This question is kind of an extension of a previous question I asked here.
The infinite series
$$\sum\frac{\mathrm{sgn}(\sin(n))}{n}$$
does converge, but I would like to know if Dirichlet's test can be used to prove the convergence…

Jaeseop Ahn

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**26**

votes

**2**answers

### Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$
Prove it converges and,
evaluate the series.
For the first part of the question, I prove it converges by considering the irrationality measure,
$$|\sin…

Tianlalu

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**25**

votes

**2**answers

### 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 measurable?

Vladimir Reshetnikov

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**24**

votes

**5**answers

### Does there exist $\ n,m\in\mathbb{N}\ $ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\ $?

Does there exist $\ n,m\in\mathbb{N}\ $ such that $\ \lvert \left(\frac{3}{2}\right)^n - 2^m \rvert < \frac{1}{4}\ $ ?
I have tried for the first few integers $\ n,m\ $ up until $\ m\approx30\ $ with no $\ n,m\ $ satisfying the inequality. However,…

Adam Rubinson

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