# Questions tagged [irrational-numbers]

An irrational number is a real number that cannot be expressed in the form $\frac{n}{m}$ where $n$ and $m$ are integers.

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### Two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number

Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?
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### An asymptotic formula in Apéry's proof of the irrationality of $\zeta(3)$

Let $a_n$ be the Apéry sequence $$a_n = \sum_{0\leq k\leq n}\binom{n}{k}^2\binom{n+k}{k}^2.$$ Reading the 1978 paper Démonstration de l’irrationalité de $\zeta(3)$ (d’après R. Apery) of Cohen, at ...
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### Irrationality of this trigonometric function

I'd like to prove the following conjecture. Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$). Then $f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$ is irrational if ...
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### Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?

Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function. If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
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### Ergodic Theory and Euler-Mascheroni Constant

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a ...
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### Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?

We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain. Is it possible to estimate ...
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### Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)

If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
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