# Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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### Approximating reals by p-smooth ratios

Is there a techinque that allows us to approximate, with arbitrary precision, any positive real number $x$ by a sequence of ratios of $p$-smooth numbers, for given prime number $p>2$? If yes, is ...
• 349
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### $|e^{inr}-1| \geq C>0$, $\forall n \in \mathbb{Z}$

I'm looking for an optimal assumption on $r$ such that $|e^{2inr}-1| \geq C>0$, for any $n$. It is clear that $$|e^{2inr}-1|^2=2(1-\cos(2nr)).$$ If $r$ is irrational, $1-\cos(2nr)>0$ but it is ...
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### How fast can a series of positive rationals converge if we know its limit is rational?

Question: Let $\{x_n\}_{n\ge 0}$ be a sequence of positive reals for which $\sum_{n\ge 0}x_n$ converges. Does there always exist a sequence $\{y_n\}_{n\ge 0}$ of rationals such that $0<y_n<x_n$ ...
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### How does Wolfram Alpha find polynomial equation of given roots?

I am experimenting with a method which will converge hopefully to a real number, for which I suspect, that it is the root of a polynomial equation. How does Wolfram Alpha find its guess? How does WA ...
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### Can $c^2(a\cdot b)+c(a+b)=2^c-2$ be solved to find $\mathbb{N}$ solutions?

$c^2(a\cdot b)+c(a+b)=2^c-2$ is of the form $mx+ny=k$ and should open the door to Diophantine, but do the constraints $x=(a\cdot b), y=(a+b)$ make a difference when trying to solve for $\mathbb{N}$ ...
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### Does $\lfloor \tan (n)\rfloor =n$ have infinite roots?

Background This post can be understood as the gap between $\tan n$ and $n$ has no upper limit. Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound? I want to know how close $\tan n$ can ...
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1 vote
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### If $\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$ then $p/q$ is necessarily one of the convergents : Extend the proof to irrational $x$

Prove that, if $x$ is any irrational number, and if $p/q$ is a rational fraction in lowest terms, with $q\geq 1$, such that $$\Big|x-\frac{p}{q}\Big|<\frac{1}{2q^2}$$ then $p/q$ is necessarily one ...
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1 vote
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### Validity of Equality in the Condition for a Rational to be a Convergent

Let $x$ be an irrational number and the rational number $𝑎/𝑏$ satisfy the inequality: $$\bigg|𝑥−\frac{a}{b}\bigg|<\frac{1}{2b^2}$$ Then $𝑎/𝑏$ is a convergent of $x$ A proof by contradiction ...
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### Rational solutions of the equation $2xy^2=kx^2+1$

I am looking for the rational solutions of the equation: $$2xy^2=kx^2+1$$ where $k$ is a fixed positional natural number. For $k=6$, Maple shows it is irreducible and doesn't produces the rational ...
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### Does every line through the origin come arbitrarily close to some other lattice point?

Given positive real numbers $x$ and $\epsilon$, do there necessarily exist positive integers $p$ and $q$ such that $\left| qx - p \right| < \epsilon$? Or geometrically: in a two-dimensional ...
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