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

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-3
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0answers
85 views

Open problems of Clay Mathematics Institute. [on hold]

Mathematics research at the end of the $20$th century assumed unfathomable proportion as compared to the situation at the end of the $19$th century. The Clay role is to further the legacy of Hilbert's ...
10
votes
2answers
157 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 ...
2
votes
0answers
66 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, ...
0
votes
0answers
32 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?
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0answers
44 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
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1answer
86 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 ...
3
votes
0answers
72 views

Big list of references [divided by categories] that collect commented open problems and conjectures [closed]

The aim of this question is to collect a big list of books or survey papers or websites which collect an up-to-date, comprehensive, well-organized, and possibly commented list of open problems. I ...
7
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3answers
346 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 ...
0
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1answer
16 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, ...
0
votes
0answers
15 views

Does Schinzel's hypthesis hold when you allow exponentials?

I'm curious to what extent Schinzel's hypothesis is expected to hold. Heuristically it seems that a typical doubly exponential function say $3^{3^n}+4$ probably is prime only at finitely many $n$ by ...
1
vote
1answer
73 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
197 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. ...
65
votes
23answers
1k views

Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both ...
6
votes
2answers
216 views

Why is there apparently a consensus on the P = NP question?

So through my years of education I have heard a lot about the famous $\mathrm{P}=\mathrm{NP}$ problem. I have seen that a significant number of mathematicians believe that this result is false (and ...
1
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0answers
28 views

Is there a finite set comprising the solutions to indefinite integrals of common functions?

There are some integrals that are impossible to express in terms of elementary function, for example, $ \int \frac{e^x}{x} dx $ is only expressible as a "special" function $Ei(x)$, the exponential ...
1
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1answer
41 views

Sum on digits of powers of two is not too large

Is the following proved: Are there infinitely many positive integers $m$ and an integer $n$ such that sum of digits of $2^m$ is at most $n$?
0
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1answer
26 views

NP-Completeness and NP

Given : $S$ is an $NP-Complete$ problem $Q$ and $R$ are two other problems not known to be in $NP$. $Q$ is polynomial-time reducible to $S$ and $S$ is polynomial-time reducible to $R$. My thoughts ...
6
votes
1answer
297 views

Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
0
votes
0answers
20 views

Dominance Network Worded Problems

What are some methods to solve this? Normally for dominance I do as such: Write a matrix for one step dominance, then find total dominance by = D+D^2 - then sum each row of the matrix. Using this ...
0
votes
1answer
54 views

Problem Solving - Project Crashing Time

My working out: (EST,EFT) times for the activities: A: (0,0) B: (0,8) C: (3,3) D: (10,38) E: (10,18) F: (18,18) G: (25,33) H: (58,58) I: (25,33) J: (45,53) K: (118,118) Finish: (133,133) ...
9
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1answer
113 views

Any book on major (recent) math discovery (results) in an easy understanding way?

All: Can anyone recommend a book which illustrate major (recent) math discoveries (results) in an easy understanding way ? For "recent discoveries", I meaning something discovered in last 50 years. ...
3
votes
1answer
98 views

What are the big issues in modern graph theory?

This is inspired by the similar question on modern set theory. I've read through the open problems in graph theory on Wikipedia's list of unsolved problems in mathematics, but what I'm looking for is ...
45
votes
7answers
5k views

What is the oldest open problem in geometry?

Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history. What I would like to know is: What is the oldest open problem in ...
3
votes
0answers
73 views

On the indivisibility of odd perfect numbers by small numbers

A good day to everyone! This question is an offshoot of the following MSE posts: Odd perfect number divisors Can an odd perfect number be divisible by $101$? My question is as follows: Is ...
1
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0answers
86 views

Research Topics Needed

This coming academic year a professor has asked me to find some topics that I wish to pursue to write about. The problem/topic that will be discussed doesn't have to be open, but my trouble is that I ...
1
vote
2answers
91 views

How can I solve this problem without having to do it by hand?

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 ...
0
votes
1answer
75 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 ...
1
vote
2answers
81 views

Is there any way to solve this problem without having to do 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. Is there any way to group ...
2
votes
0answers
48 views

What do we know about Sorli's Conjecture and Odd Perfect Numbers?

I have seen that Jose Arnaldo Dris has recently published a great deal of literature concerning odd perfect numbers and Sorli's Conjecture. He claims that the truth of Sorli's conjecture implies that ...
7
votes
1answer
211 views

Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
2
votes
2answers
114 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
13
votes
2answers
569 views

Are there any infinite sets that are not known to be either countable or uncountable?

Are there any known examples of sets that are definitely infinite, but where we don't know whether or not they're countable? I haven't heard of anything like this before, but it seems that there ...
1
vote
0answers
26 views

Why the proof of Catalan's conjecture is not easily generalizable?

Let $x,y>0$, $u,v>1$ be integers. Why is it easier to solve $x^u-y^v=1$ than $x^u-y^v=2$? Is there possible some group behind the first equation which has some nice property that the group made ...
2
votes
2answers
97 views

State of art of prime numbers distribution [closed]

I was reading some questions about prime numbers posted in latest days and a question came to my mind: What is the state of art of the research into prime numbers distribution? I read then ...
0
votes
0answers
19 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 ...
2
votes
1answer
181 views

Twin Prime conjecture current status

Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin ...
1
vote
1answer
83 views

Can one prove a special case of Goldbach conjecture without constructing primes?

I know that sometimes in mathematics one can prove that there exist something without constructing it. I was thinking whether one can show if $2^{57885162}$ is a sum of two primes by any reasoning. ...
2
votes
0answers
97 views

A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
0
votes
1answer
241 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 ...
0
votes
0answers
75 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 ...
4
votes
1answer
271 views

Can the twin prime conjecture be solved in this way?

After some research, I have discovered that proving the statement; There exist an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a \pm b$ is ...
1
vote
1answer
50 views

Does proving the following statement equate to proving the twin prime conjecture?

After some research, I found that it has been supposedly proven, that proving that there exists an infinite number of positive integers K such that; $K \neq 6ab \pm a \pm b$ and $K \neq 6ab \mp a ...
16
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1answer
602 views

What are some of the major open problems in category theory?

What are some of the major open problems in category theory? Just curious - I'm interested in category theory.
0
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1answer
93 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 ...
6
votes
1answer
162 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
0
votes
1answer
97 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 ...
4
votes
2answers
171 views

Which precision would be needed?

According to Wikipedia, it is not known whether the number $$\pi \uparrow \uparrow 4$$ is an integer. (See Tetration) To which precision would $\pi$ have to be calculated to decide this ? The ...
2
votes
1answer
283 views

Is there any hope to disprove Goldbach's conjecture?

It is widely believed, that Goldbach's conjecture is true. But suppose, there is a counterexample of, lets say, 50 digits. Is there any hope to prove this counterexample to be one ? Brute force ...
4
votes
0answers
86 views

A light beam enters a closed room. What is the maximal number of reflections?

I have the following problem: a light beam enters a mirror room with integer coordinates in the plane (consider it as a polygon). One of the walls of the room is removed and the light beam enters the ...
1
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0answers
71 views

$q\sin q$ is small

I read from the book "Which Way did the Bicycle Go" that it is unknown whether for every $c>0$ there are infinitely many integers $n$ such that $|n\sin n|<c.$ Let $\mathbb{Q}_{m}$ be the set ...