For questions about approximating real numbers by rational numbers.

# Questions tagged [diophantine-approximation]

432 questions

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### show that $\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$…

math110

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### Find the solutions to $\left\lfloor\left(\frac{5}{3} \right)^n\right\rfloor = 3^m$

Find all positive integer solutions to $\left\lfloor\left(\dfrac{5}{3}
\right)^n\right\rfloor = 3^m$.
Let $a_n = \left\lfloor\left(\dfrac{5}{3}
\right)^n\right\rfloor$. Then $$a_n =…

user19405892

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### Prove the inequality $\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$

Prove that the inequality $$\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$$ holds for positive integers $m, n$ if and only if $m$ and $n$ where $m > n$ are two successive terms of the Fibonacci sequence.
I thought about using the…

user19405892

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### Proving $\left|\sqrt2-(a/b)\right|\geq1/(3b^2)$

This is Problem 4.26 on p.58 of The Theory of Algebraic Numbers by Harry Pollard and Harold G. Diamond (Dover edition).
Prove that $\left|\sqrt2-\dfrac ab\right|\geq\dfrac1{3b^2}$ for all positive integers $a,b$.
Here's what I've done so far.…

George Law

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### 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 community standard; if you know, please let me…

Bobson Dugnutt

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### Floor function inequality $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$

Let $a,b$ be positive integers. Show that $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$.
[Source: Russian competition problem]

simmons

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### 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 with high probability. I wonder if it can be…

Tanny Libman

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### Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?

Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$?
Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it still holds?
Numerical experiment.
By plotting…

Kemono Chen

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votes

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### Solutions to the diophantine equation $x^n-2y^n=1$. Can the sum of the first $n$ squares be a perfect power?

This is an attempt to generalize this question. $$x^n-2y^n=1\implies \frac{x}{y}=\left(2+\frac{1}{y^n}\right)^{\frac 1 n}=2^\frac1n\left(1+\frac{1}{2y^n}\right)^\frac1n<2^\frac…

Sophie

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### 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 logarithms because two transcendental constants are…

Jaume Oliver Lafont

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### 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 (a,b) \in {\cal L} \bigg| \forall (a',b') \in {\cal…

Ewan Delanoy

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votes

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### 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 integer?
For example: $\sqrt[3]{100}+\sqrt[3]{4}$…

Bart Michels

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### Inequality $|\cos(k)| \geq \frac{1}{2^k}$ for $k\geq 0$

My question : Is it true that $|\cos(k)| \geq \frac{1}{2^k}$ for all integers $k\geq 0$ ?
What I tried : I have checked with a computer that the inequality holds for
$0 \leq k \leq 4\times 10^5$. I can also show that the set of $k$ for which the…

Ewan Delanoy

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### 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 $2$. $( 5^3 + 5 + 6^3 + 6 = 7^3 + 7 + 2 )$.
$(6,8,9)$ is true…

mick

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### Irrationality measure of $\alpha+\beta$, $\alpha\beta$

Let $\alpha$ and $\beta$ be real numbers with finite irrationality measures. My question is:
Are the irrationality measures of $\alpha+\beta$ and $\alpha\beta$ also finite?
I tried using triangle inequality
$$
\left| \alpha - \frac {p_1}{q_1} +…

Sungjin Kim

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