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

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Legendre's Conjecture Theme (Part II)

This is a continuation of this question. My main question is that, in the previous question we were mainly concerned about the sign of, $$f_{2}(n)=\pi\left((n+1)^2\right)+\pi\left(n^2\right)-2\pi\left(...
2
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0answers
65 views

Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
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0answers
22 views

Problem of rectilinear motion [closed]

$a=-{\dfrac{C}{s^2}}$ where $C=gr^2$. Neglect all resistance.(a) Now,let a body start from rest at a distance $h$ from the surface of the earth (radius r). Choose the centre of the earth as origin. ...
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0answers
15 views

Unsolved problems involving or concerning the unitary DFT

Does anyone know of any unsolved problems involving or concerning the unitary discrete Fourier transform matrix $F_n=n^{-1/2}(f_{j k})$ where $f_{jk }=e^{2\pi j k i}$ and $i=\sqrt{-1}$, or its inverse?...
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1answer
70 views

A question about lonely runner conjecture

After wikipedia: "Consider $k$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is ...
2
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1answer
71 views

Open problems in Galois theory (other than the IGP)

I'm interested in open problems in Galois theory. It's not necessary for them to be well known or considered important, but they have to be mostly Galois-theoretic, that is, not number theory or ...
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0answers
15 views

On the density of solitary numbers

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. If $X$ is the unique solution of $$I(X) = \dfrac{a}{b}$$ (for a given rational ...
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1answer
48 views

Why is it called the *Inverse* Galois Problem?

This is just a very quick question and hopefully not poorly received. Question: Why is it called the inverse galois problem? The very brief statement given on wikipedia says Is every finite ...
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0answers
313 views

Is this proof of the twin prime conjecture? [closed]

Identifying twin primes [1] Any natural number $n : 1<n\leq p_x^2 $ where $n$ is not divisible by any prime number less than $p_x$ is a prime number, except when $n$ is one of those prime ...
13
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1answer
129 views

Are derivatives of geometric progressions all irreducible?

Consider the polynomials $P_n(x)=1+2x+3x^2+\dots+nx^{n-1}$. Problem A5 in 2014 Putnam competition was to prove that these polynomials are pairwise relatively prime. In the solution sheet there is the ...
2
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0answers
40 views

Existence of a cycle of length $2^k$?

I was given a question which asks me to show that if each vertex of a graph has degree$\geq 3$, then it has a cycle whose length is some power of $2$. I have been able to show that it has a even ...
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10 views

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, which of the following sets of inequalities *cannot* hold?

Let $\sigma(X)$ be the sum of the divisors of $X$. If $\sigma(Y) = 2Y$, then $Y$ is said to be perfect. If $N$ is odd and perfect, then $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ ...
4
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1answer
110 views

Is this really an open problem? Maximizing angle between $n$ vectors

It is well known that the trigonal planar molecule (with bond angle $\alpha=120^{\circ}$) and the famous tetrahedral (with bond angle $\alpha\approx 109.5^{\circ}$) maximizes the angle between the ...
14
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1answer
208 views

Is $0.248163264128…$ a transcendental number?

My question is in the title: Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$). It is possible to express $a$ as a ...
25
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1answer
350 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
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0answers
54 views

Estimates for the Dedekind number $M(9)$

The Dedekind number $M(n)$ is the number of antichains in the partial order of subsets of $\{1,\dotsc,n\}$. It is only known for $0 \leq n \leq 8$. Question. What are some known upper and lower ...
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1answer
27 views

Under what conditions is the implication $I(x) < I(y) \implies x < y$ true?

Let $\sigma(x)$ be the sum of the divisors of $x$, and denote the abundancy index of $x$ by $$I(x) = \frac{\sigma(x)}{x}.$$ My question is: Under what conditions on $x$ and $y$ is the implication ...
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4answers
27 views

Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers

(Note: This question has been cross-posted from MO.) Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) =...
2
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1answer
29 views

Measure of convex hulls

I'm not an expert of this kind of questions, but I can't give a satisfactory answer to the following question. Pick $x_1\dots x_n \in \mathbb{R}^m$. Is there a formula for the measure of the ...
106
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19answers
4k views

Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved ...
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1answer
446 views

The four runner problem/conjecture

I've recently read here the following problem, called « four-runner problem » : Suppose four runners (represented by labeled spheres) run around a circular track. Their speeds are constant ...
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0answers
41 views

Covering unit square

Now, I am reading this topic http://mathoverflow.net/questions/34145/can-we-cover-the-unit-square-by-these-rectangles. And do some research on it. Guys, who had written in topics, have said, that they ...
0
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0answers
31 views

Can distinct odd perfect numbers $N = {p^k}{m^2}$ share the same Euler factor $p^k$?

(A similar question has been asked in MO.) Let $\sigma(x)$ denote the sum of the divisors of $x$, and call the ratio $I(x) = \sigma(x)/x$ as the abundancy index of $x$. A number $N$ is called ...
11
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1answer
3k views

Fractional oblongs in unit square via the Paulhus packing technique

Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, ... have a total area of 1. $\sum\limits_{...
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3answers
142 views

Is it possible for extremely ingenious but elementary proofs for famous problems to exist?

As Erdős put it, "Mathematics is not ready for such problems." when faced with the great conjecture of Collatz. So is it impossible altogether for simple but ingenious proofs for famous problems ...
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0answers
82 views

Up to which value is Rassias' conjecture verified?

I came across this conjecture: Rassias' conjecture Up to which $p$ has this conjecture be verified ? Are there intermediate results related to this conjecture ? The conjecture can be ...
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1answer
32 views

Integers $n\geq 2$ such that there is an integer $k>1$ that divides both $n$ and $(n/k)+1$

I apologize to edit this question to request suggestions in this case for study on a sequence starting, $n\geq 2$ such that there is an integer $k>1$ that divides both $n$ and $(n/k)+1$, because it ...
1
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1answer
96 views

What does the Grothendieck's period conjecture mean?

I would like to know how is the Grothendieck's period conjecture about algebraic cycles, defined explicitly ? and, what link has it with the Hodge conjecture for smooth complexe algebraic projective ...
46
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18answers
6k views

What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It ...
4
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1answer
75 views

Infinite torsion CAT(0) groups

Do all infinite CAT(0) groups contain a $\mathbb{Z}$ subgroup? I am aware that this has been established for hyperbolic groups, and similar questions have appeared on open questions lists for CAT(0) ...
4
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3answers
211 views

Solve $x^5 + y^5 + z^5 = 2015$

If $x, y, z$ are integer numbers, solve: $$x^5 + y^5 + z^5 = 2015$$ A friend of mine claims there is no known solution, and, at the same time, there is no proof that there is no solution, but I do ...
12
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0answers
127 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
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1answer
413 views

Would this proof strategy work for proving the lonely runner conjecture?

The problem is the lonely runner conjecture. This conjecture states that if $k$ runners begin running down a circle of unit circumference with random speeds, it will always the case that all runners ...
8
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1answer
112 views

Is $2^{\sqrt{2}} +3^{\sqrt{3}}$ rational?

In an exercise found in Lang's A Complete Course in Calculus, it's mentioned that it is unknown if $2^{\sqrt{2}} +3^{\sqrt{3}}$ is rational. Has this problem been solved since then?
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1answer
40 views

4-Color Theorem question - is the set of 4-vertex-colorings of a planar graph closed under Kempe switching?

A $4$-vertex-colored planar graph $G$ is a planar graph $G \overset{\text{def}}{=} (V, E, C)$ where $V$ and $E$ are as usual and $C$ consists of pairs $(v \in V, c \in \{1,\dots,4\})$ such that $\...
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0answers
100 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
2
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1answer
54 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...
4
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1answer
237 views

How can you confirm that a problem is open?

I was reading an article on Wikipedia and I came across a list of two problems which they asserted to be open, but without citation. I have looked through some literature but not all, as I am afraid I ...
4
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1answer
87 views

Are there simple unsolved problems in statistics?

In number theory, calculus and various fields of mathematics, there are many unsolved problems. But are there simple unsolved problems in statistics?
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2answers
169 views

Would a quantum computer solve the Riemann hypothesis? [closed]

I heard that a quantum computer can give many results as one computation step. Does it mean that it would be just a brute force search for a quantum computer to solve for example the Riemann's ...
2
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1answer
82 views

Groups $\pi$ with $K(\pi, 1)$ a finite CW-complex

For what (say, finitely generated) groups $\pi$ is $K(\pi, 1)$ a finite CW-complex? Such a group must necessarily be torsion-free, since otherwise $H^*K(\pi, 1) = H^* \pi$ would be nontrivial in ...
3
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1answer
97 views

Is it possible to show that a particular theorem or its negation is provable, without knowing which of the two is true?

I've been thinking about this for a while: as far as we know, is it possible that for a particular statement $\sigma$ of $\textsf{ZFC}$, we can prove that $(\textsf{ZFC} \vdash \sigma) \vee (\textsf{...
13
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2answers
184 views

Prove that $\lim_{n\rightarrow \infty} \frac{\log_{10}\lfloor\text{Denominator of } H_{10^n}\rfloor+1 }{10^n}=\log_{10} e$

In short, my question is asking to prove that the $$\lim_{n\to\infty}\frac{\text{number of digits in the denominator of} \sum_{k=1}^{10^n} \frac 1k}{10^n}=\log_{10} e$$ I know that the number of ...
11
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2answers
660 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
3
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3answers
1k views

Result of solving an unsolved problem?

I was mulling over currently unsolved problems in mathematics (as I, and many others, find them quite interesting) and began to wonder what would happen if these problems were to be solved. I know ...
3
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0answers
69 views

Open problems in Banach spaces? universal spaces

I have gathered a list of universality problems in Banach spaces which have been solved: The non existence of a separable reflexive space universal for the class of separable reflexive spaces. If a ...
5
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2answers
162 views

Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
0
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2answers
101 views

Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link below....
5
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0answers
79 views

Weakening Goldbach hypothesis by allowing finitely many composites and 1

Is it an open question whether there is a finite set $N$ of positive integers such that for every positive even integer $n$ there are $n_1,n_2\in\mathbb P\cup N$ such that $n=n_1+n_2$? ($\mathbb P$ ...
5
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2answers
325 views

Are all $\delta$-hyperbolic groups CAT(0)?

In Alessandro Sisto's notes on geometric group theory he mentions that "Many, probably most people in the field" believe that not all $\delta$-hyperbolic groups are CAT(0) groups. Can anything be said ...