# Tagged Questions

For questions about approximating real numbers by rational numbers.

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### 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 ...
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### Diophantine approximation and covers

Suppose $\alpha > 2$. Let F be the set of real numbers $x \in [0,1]$ for which the inequality $||qx|| \le q^{1-\alpha}$ is satisified by infinitely many positive integers q. For each q, let $G_q$ ...
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### What does $\log^3$ stand for in this paper by K. Győry?

In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$. Namely, consider a ...
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### Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
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### 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 ...
In Rudin's Principles of Mathematical Analysis (3rd ed), he proves (at the very beginning: example 1.1) that the set $A$ of all positive rationals $p$ such that $p^2<2$ contains no largest number ...