# Questions tagged [diophantine-approximation]

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### Does the equation $a^b+b^c+c^a=d^e$ have solutions in $\mathbb {N}$

Here $a,b,c,d,e$ are distinct and all greater than $1$. This question was formerly posted on Math.Stackexchange, precisely here, but seems to be more general than some other tough number theory ...
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### A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
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### A pathological (?) function involving powers

This is inspired by a recent math.SE question. Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide ...
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### Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$

EDIT (August 9, 2021): I would like to ask a more general question. The original question that was fully answered is below the line. For a positive real number $x$, denote the fractional part $x-[x]$ ...
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### Given $2^n - 1 \mid 3^m - 1$, how large must $m$ be compared to $n$?

Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ ...
114 views