# Questions tagged [open-problem]

Questions on problems that have yet to be completely solved by current mathematical methods.

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### T/F: If $\alpha\in\mathbb{R}_{>1}$ is not Pisot, then $\{\lfloor\alpha^n\rfloor:n\in\mathbb{N}\}$ contains infinitely many odd and even integers

True or false: If $\alpha\in\mathbb{R}_{>1}\setminus\mathbb{N},$ then the set $\{ \lfloor \alpha^n \rfloor: n\in\mathbb{N} \}$ contains infinitely many even integers and infinitely many odd ...
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### Collatz sequence but multiplying by a large odd number rather than $3.$ What is the simplest way to prove that a sequence goes off to infinity?

Under the Collatz rules: $n\to 837n+1$ if $n$ is odd $n\to n/2$ if $n$ is even. What is the simplest argument/proof to show that there is a Collatz sequence with a starting number that goes off to ...
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### The $27$ dots problem

Here is a generalization of the well-known Nine dots puzzle to $3$ dimensions, where we introduce a new constraint (it is just a particular case of a more general problem that I have recently shared ...
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1 vote
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### What's The Minimum Number Of Prime Factors Needed To Replace "3x+1" With Any Linear ("mx+b") Function And Still Work Like The Collatz Conjecture?

Apologies; I know there are a few assumptions used to pose this question, namely: 1): That yes, any mx+b function can work like the infamous "3x+1," problem... ...Provided, that you give it ...
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### Implications of having access to the Busy Beaver oracle

Apologies if I'm asking a naïve question as I've only recently learned about the concept. What would be the practical implications (if any) of having access to a magical black box providing the ...
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### Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right)$

Background The Norwegian mathematician and astronomer Carl Størmer did important work on the equation $$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1}$$ ...
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### Can we really be sure that there is no odd perfect number below $10^{3000}$?

A positive integer $N$ is called perfect if the sum of its divisors (including $1$ and $N$) is $2N$. A famous open problem is whether there is an odd perfect number. Can someone confirm the following ...
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### Liouville Lambda Function and Riemann Hypothesis

What is the exact statement involving the Liouville Lambda function, which is equivalent to Riemann Hypothesis, and true iff RH is true? Can anyone cite the sources for it and/or outline its proof in ...
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### Can we come up with a disjoint union of a subsets of the group $\Bbb{Z}$ such that they do not equal the cosets of a subgroup, yet they form a group?

If this applies to $\Bbb{Z}$ it probably will work for other groups $G$, however, for simplicity and because I'm interested in integers & their primes, let's work with $G = \Bbb{Z}$. Anyway, we ...
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### Doubt about approaching problem from the Kourovka notebook

Problem 20.58 from the Kourovka notebook, particularly the first part, reads Is it true that for every positive integer $n$ there is a recognizable group that is the $n$-th direct power of a non-...
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(Note: This was cross-posted from MO, because it was not well-received there. Will delete the MO post in a few.) Observe that an even perfect number $M = (2^p - 1)\cdot{2^{p - 1}}$ and an odd perfect ...