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

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Proving that $\sum_{k=n}^{2n-2} \frac{|\sin k\ |}{k} < 0.7\ln 2$ for $n\ge2$

Prove that $$\sum_{k=n}^{2n-2} \frac{|\sin k|}{k} < 0.7 \ln 2, \qquad (n\ge2)$$ and $$\cot\left(\frac{\pi}{2n}\right) \le \sum_{k=1}^n \left|\sin\left(x+\frac{k\pi}{n}\right)\right| \le ...
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1answer
232 views

Waring's Inequality Solution

$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$ We craft an inequality, with $m,n ...
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1answer
124 views

Catch stochasticity of nature

Do you know of anything that comes close to topic 3 on http://www.darpa.mil/Our_Work/DSO/Programs/23_Mathematical_Challenges.aspx Capture and Harness Stochasticity in Nature Address Mumford’s ...
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1answer
286 views

symmetric difference of languages - both are in NP and coNP

I have this problem, Let $L_1,L_2$ be languages in $NP \cap co-NP$. I want to show that their symmetric difference is also in $NP \cap co-NP$. Like: $L_1 \oplus L_2 = \{x | x$ is in exactly one of ...
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1answer
68 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
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0answers
12 views

an idea about the relation between the residues at infinity and the infinity line?

i have a question about the following three concepts for infinity: The point at infinity, also called ideal point, of the real number line is a point which, when added to the number line yields a ...
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64 views

Conjecture related to the Erdős discrepancy problem

Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to ...
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1answer
59 views

Perfect cuboid cube

Is there any proof that there is no cubic perfect cuboid? Here is a description of the problem: . I'm currently using trying to get an empty set to solve it... [ A "perfect cuboid" is one whose ...
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0answers
83 views

When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ are ...
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236 views

Possible proof for Frankl's Union-Closed Sets Conjecture?

Frankl's union-closed sets conjecture: if $F$ is a nonempty finite collection of nonempty finite sets, and if $X\cup Y\in F$ whenever $X,Y\in F$, must there be an element which is in more than half ...
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1answer
58 views

will $x_{n+1}=x_n/2$ if $x_n$ is even; otherwise $x_{n+1}=3*x_n+1$, will $x_n$ shrink to 1?

I was asked this question that, for any $x_1 \in \mathbb{N}$, define the sequence as $$x_{n+1}=\left\{ \begin{array}{l l} x_n/2 & \quad \text{if } x \text{ is even} \\ 3 x_n+1 & ...
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0answers
108 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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51 views

Reference for the cardinality equal to continuum of space

Recently, I am considering an open problem, see here. Now, I want to prove the answer to the problem is true when the case that the cardinality of topological space is equal to continuum. However, I ...
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1answer
89 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
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2answers
127 views

Has the OPN Conjecture been finally proved? [closed]

Has the OPN Conjecture been finally proved? All perfect numbers are even, by Sanjit Singh Batra and Amitabha Tripathi
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2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
-5
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1answer
459 views

Have 3n + 1 Problem Proof --But can't do the Mathspeak [closed]

I've come here as a last resort. When I first saw the Collatz conjecture I worked out pretty quickly what the dynamic was that drove the Collatz sequences. That was in 2009. Since then I've been ...