# All Questions

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### Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Sorting of prime gaps

Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we rearrange the sequence $(g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new ...
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### Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
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### Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
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### Evaluating $\sum\limits_{x=2}^\infty \frac{1}{!x}$ in exact form.

Introduction: We know that: $$\sum_{x=0}^\infty \frac{1}{x!}=e$$ But what if we replaced $x!$ with $!x$ also called the subfactorial function also called the $x$th derangement number? This ...
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### This should be a piece of cake...... right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut ...
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### Grasshopper jumping on circles

Can we characterize the grasshopper sequence? Let $n\in\mathbb N$ be the number of stones $s\in\{0,1,2\dots,n-1\}=S$ on a circle that the grasshopper can jump on. Let $v(s)$ be the number of times ...
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### Is every finite list of integers coprime to $n$ congruent $\pmod n$ to a list of consecutive primes?

For example the list $(2, 1, 2, 1)$ is congruent $\pmod 3$ to the consecutive primes $(5, 7, 11, 13)$. But how about the list $(1,1,1,1,1,1,1,1,2,3,4,3,2,3,1) \mod 5$? More generally, we are given ...
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### Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
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### Does any uncountable complete theory have exactly two countable models?

The following is a theorem by Vaught. Theorem. Let $T$ be a complete theory in a countable language. Then, $T$ cannot have exactly two countably infinite models (up to isomorphism). A proof can ...
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