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

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18
votes
1answer
120 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 ...
1
vote
0answers
46 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 ...
0
votes
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 ...
-2
votes
4answers
20 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
votes
1answer
27 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 ...
98
votes
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 ...
14
votes
1answer
375 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 ...
1
vote
0answers
36 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
votes
0answers
27 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
votes
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. ...
4
votes
3answers
127 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 ...
3
votes
0answers
62 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 ...
1
vote
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
vote
1answer
76 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 ...
45
votes
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
votes
1answer
56 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
votes
3answers
204 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 ...
10
votes
0answers
107 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 ...
1
vote
1answer
248 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
votes
1answer
107 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?
1
vote
1answer
38 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 ...
5
votes
0answers
77 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
votes
1answer
51 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
votes
1answer
227 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
votes
1answer
75 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?
-2
votes
2answers
134 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
votes
1answer
74 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
votes
1answer
90 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 ...
13
votes
2answers
182 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 ...
9
votes
1answer
421 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
votes
3answers
896 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
votes
0answers
63 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
votes
2answers
141 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
votes
2answers
83 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 ...
0
votes
0answers
28 views

A detail in [ the time evolution operator]

If $$\hat C=\Delta \hat S_{22}+\lambda _1[\hat a_1 \hat S_{21}+\hat a_1^\dagger \hat S_{12}]+\lambda _2[\hat a_2 \hat S_{32}+\hat a_2^\dagger \hat S_{23}] ,$$ $$ \hat \beta= ...
5
votes
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 $n=n_1+n_2$? ($\mathbb P$ ...
0
votes
0answers
39 views

diffusion- stuck

In a round room of radius R, a large number of coins N of diameter d are randomly dispersed upon the floor. A ladybird starts from the centre of the room, crawling at speed v. Suppose that every time ...
5
votes
2answers
244 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 ...
3
votes
0answers
37 views

<Reference Request> Research done on whether the Euler prime can be the largest factor of an odd perfect number

(Note: This has been cross-posted to MO.) Good day! I would like to request for references to research done as to whether the Euler prime of an odd perfect number can also be its largest factor. ...
10
votes
2answers
184 views

If $3^x$ and $5^x$ are both integers, is $x$ an integer?

Does the following statement hold? $$x\in \mathbb{R}^+ \text{and} \ 3^x, 5^x \in \mathbb{Z} \implies x \in \mathbb{Z}$$ In words: If $x>0$ is a real number, and $3^x$ and $5^x$ are ...
6
votes
1answer
454 views

Innocent looking open problems in real analysis

Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, ...
1
vote
0answers
54 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
1
vote
0answers
65 views

Open problems for which all cases except one have been solved

Keller's conjecture states that in any tiling of Euclidean $n$-space by identical hypercubes there are two cubes that meet face to face. The conjecture has been shown to be true for $n<7$ and ...
2
votes
1answer
101 views

two questions about primes

I'm very ignorant about results in number theory concerning the primes. Please let me know if these are open conjectures or easy problems: There are infinitely many primes of the form $n!+1$ There ...
7
votes
3answers
419 views

On progress in mathematics: some long-open problems and long-standing conjectures

I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1) Not especially famous, long-open problems which ...
3
votes
0answers
85 views

Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
0
votes
1answer
30 views

Dickson's (and Bunyakovsky's) conjecture with compositeness constraints

Dickson's conjecture, in simple terms, says that for any choice of $a_1,b_1,a_2,b_2,...,a_k,b_k\in\Bbb N$ we have, for infinitely many $n\in\Bbb N$, that all of $a_1+nb_1,...,a_k+nb_k$ are prime, ...
4
votes
2answers
319 views

Is it known if $\pi + e$ is transcendental over the rational numbers?

I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as ...
2
votes
1answer
98 views

Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This has been cross-posted to MO.) Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$). ...
10
votes
3answers
224 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...