Questions tagged [open-problem]

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

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Questions about density of $\left\{\ \left( k+ \frac{1}{2} \right)^n \right\}$ in $[0,1]$

Here, $\{ x \}$ denotes the fractional part of $x.$ Are there any known positive integers $k$ for which the set $\left\{\ \left\{ \left( k+ \frac{1}{2} \right)^n \right\}: n\in\mathbb{N}\ \right\} $ ...
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-1 votes
0 answers
38 views

How relevant would it be to prove that P vs NP is equivalent to P vs NP using only machines with one letter input alphabet?

I was reading the official description of P vs NP at https://www.claymath.org/sites/default/files/pvsnp.pdf out of curiosity and the authot says "Does $\textbf{P = NP}$? It is easy to see that ...
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2 votes
3 answers
372 views

Why is it so difficult to prove $\mathbf{NP} ≠ \mathbf{P}$?

I'm just curious because I saw on Wikipedia a single polynomial time solution to any NP-hard problem would imply there are polynomial time solutions for every single NP problem. Also I assume there ...
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2 votes
1 answer
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What makes distributed function computation difficult?

Encoder $k$ informs a decoder of a source $X_k$ at rate $R_k$; $k=1,\dots, K$. The decoder seeks to recover $X_0=f(X_1,\dots, X_K)$ with high probability. Do schemes exist with failure rate ...
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11 votes
0 answers
484 views

For positive $a$, $b$, $c$, $d$, if $\sum_{cyc}\frac1{1+a}=2$, (dis)prove $\frac{4(3\sqrt2-4)}{a+b+c+d}+\sum_{cyc}\frac1{\sqrt a}\geq3\sqrt2$

An open problem from Art of Problem Solving (AoPS): If $a,b,c,d$ are positive real numbers such that $$\frac{1}{1+a}+\frac 1{1+b}+\frac 1{1+c}+\frac 1{1+d}=2,$$then prove or disprove $$\frac1{\sqrt a}...
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0 votes
2 answers
170 views

If there are $k_1, k_2, \dots, k_n$ divisions by $2$ in a Collatz cycle, then $k_1 + \dots + k_n \geq n$, but can we get a greater lower bound?

Let $f(x) = |3x + 1|_2(3x + 1)$ be the accelerated Collatz function, where $|3x + 1| = 2^{-\nu_2(3x + 1)}$ is the $2$-adic absolute value. Clearly for all $x$ odd we have $\nu_2(3x + 1) \geq 1$ so ...
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7 votes
1 answer
169 views

The Collatz Conjecture function should induce a collection of Grothendieck groups, one for each $n \in \Bbb{Z}$ or $\Bbb{N}$. Their properties?

This question is about the Collatz conjecture. Let $\Bbb{N}$ include $0$. The Collatz conjecture function is given by: $$ f: \Bbb{N} \to \Bbb{N}, \\ f(n) = \begin{cases} \dfrac{n}{2}, \text{ if } n = ...
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2 votes
1 answer
123 views

What is wrong with the following proof that $\{ (\frac{3}{2})^n\mod 1: n\in\mathbb{N} \} $ is dense in $\ [0,1]\ $?

What is wrong with the following straightforward proof that $\ \lbrace{ \left(\frac{3}{2}\right)^n \mod 1: n\in\mathbb{N} \rbrace}\ $ is dense in $\ [0,1]\ $ ? In fact, let $\ f(x) = \lbrace{\ x^n \...
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1 vote
0 answers
131 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part VI

(Note: This question has been cross-posted to MO.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\...
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1 vote
0 answers
56 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part V

(Preamble: This post is an offshoot of this MSE question and this MO question.) My primary aim in this post is to compute a (hopefully factorable) expression for the quantity $n^2 - q^k$, if $N = q^k ...
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1 answer
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On a possible proof for $p^k < m$, if $p^k m^2$ is an odd perfect number with special prime $p$

Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of divisors of the positive integer $x$. If $\sigma(M)=2M$ and $M$ is odd, then $M$ is called an odd perfect number (hereinafter abbreviated as OPN)...
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0 votes
2 answers
269 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part IV

(Preamble: This post is an offshoot of this earlier MSE question.) The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\...
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2 votes
2 answers
191 views

what is so special about the number $3$ in the $3n+1$ conjecture?

Recently, I was quite intrigued by the $3n+1$ conjecture and it left me wondering what is so special about the number $3$? With the same rules apply, does there exist another positive integer other ...
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3 votes
0 answers
119 views

Open problems in Proof theory and Logic

There are numerous questions in the same form: "What are some open problems in mathematical logic". So for this we know: Shelahs "Logical Dreams" Logical Dreams Friedmans "102 ...
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0 votes
1 answer
99 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part III

Preamble: This post is an offshoot of this earlier MSE question. The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\...
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1 vote
1 answer
188 views

What makes an unsolved problem "interesting"? [closed]

Beyond well-known unsolved problems like the Collatz conjecture or Recaman's sequence, one can trivially come up with problems in a similar vein (or perhaps an unusual infinite sum, or the ...
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Does this "improvement" to $I(m) < 2$, if $p^k m^2$ is an odd perfect number with special prime $p$, work?

(Preamble: This question is an offshoot of this earlier post.) Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, and the abundancy index of $x$ by $I(x)=\...
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1 vote
1 answer
417 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

The topic of odd perfect numbers likely needs no introduction. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. If $n$ is odd and $\sigma(n)=2n$, then we ...
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3 votes
0 answers
222 views

On odd perfect numbers $p^k m^2$ with special prime $p$, satisfying $m^2 - p^k = 2^r t$

In what follows, we will denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A positive integer $y$ ...
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1 vote
1 answer
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Three dimensional Hadwiger Nelson Problem

I am interested in Hadwiger Nelson Problem in higher dimensions. In particular, I have seen that the chromatic number for the Hadwiger Nelson Problem in three dimensions is between 6 and 15. But I ...
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1 vote
1 answer
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Hadwiger-Nelson problem only on Q rationals

In the Hadwiger-Nelson problem, any two points unit distance apart must have distinct colors. However, it is known that if we restrict the vertices to only rational numbers, the chromatic number is ...
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2 votes
2 answers
42 views

Frankl for infinite set

Let $A$ be a set and $F\subset \mathcal P(A)$, and for any $a\in A$, let's define $F_a:= \left\{X\in F, a\in X\right\}$ Suppose that : $F$ is not a finite union of chains for any $E\subset F$, $\...
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  • 311
2 votes
1 answer
59 views

Frankl Conjecture for infinite set

Let $A$ be a set and $F\subset \mathcal P(A)$, and for any $a\in A$, let's define $F_a:= \left\{X\in F, a\in X\right\}$ We suppose that 1)There is no chain $C\in F$ such that $\bigcup (F-C)\subset \...
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  • 311
0 votes
1 answer
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Longest-ever time between problem posing and solving?

Guiness World Record claims that Goldbach's conjecture is the oldest unsolved problem. A natural related question is what solved problem went unsolved for the longest time. In other words, of all the ...
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5 votes
0 answers
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Existence of continued fraction $\sqrt{n}$ with any period $k$

In this paper it is conjectured that for any positive integer $k$ there are infinitely many primes $p$ with the continued fraction expansion of $\sqrt{p}$ having length $k$ (Conjecture 5.1, https://...
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2 votes
0 answers
47 views

Small question about the Lonely runner conjecture

I am currently looking into the lonely runner conjecture (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' ...
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2 votes
1 answer
114 views

A combinatorial sieve for a specific setup

Let $F\subset\mathbb{Z}$ be a sifting set composed of $n\geqslant 2$ pairs of residue classes defined as $$F=\bigcup\nolimits ^{n}_{i=1}\{x:x\equiv \pm m_{i} \pmod{p_{i}}\}$$ where $p_{i}$ are ...
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2 votes
1 answer
166 views

Is it a known result that the Collatz Tree repeats on itself indefinitely?

With reference to my visual pattern of the Collatz problem, I recognize that the Collatz tree for numbers of the form $6x-1$ (for $x\in\mathbb{Z}$ & $x>0$) and $1-6x$ (for $x\in\mathbb{Z}$ &...
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  • 634
8 votes
1 answer
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Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics. Several attempts have been made in the last 150 years (here some of them are reported). RH ...
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  • 2,249
1 vote
0 answers
81 views

Open questions in algebraic geometry that need insights from computations

I think that everything is in the question. I am looking for any open problems in algebraic geometry (vector bundles, Divisors, Frobenius morphism, counting points on Finite fields, Hitchin Fibration, ...
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  • 2,730
0 votes
2 answers
128 views

Has Any Currently Open Problem in Mathematics Definitively Been Shown to be Decidable?

There is a fairly extensive list of problems in various fields that have been shown to be undecidable. For example, see https://en.wikipedia.org/wiki/List_of_undecidable_problems And certainly, an ...
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2 votes
1 answer
119 views

On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part II

(This question is an offshoot of this earlier post.) Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the sum of the ...
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2 votes
0 answers
76 views

A Question Regarding Euler's Phi Function and Composite $N$

Let $N$ be a composite number. Do we know any necessary conditions on $N$ that will potentially allow $\phi(N) \mid N - 1$? Thank you.
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3 votes
1 answer
89 views

Looking for (Previously Open) Problems That Have Been Resolved Contrary to Expectation

I am looking for reasonably famous mathematical problems that were once open for more than twenty five years (preferably more than a hundred) and whose subsequent resolution (proof or counterexample) ...
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2 votes
3 answers
213 views

$u_{n+1}=\frac{e^{u_n}}{n+1}$

One can prove that for $x\in \mathbb{R}$, the sequence $$ u_0=x\text{ and } \forall n\in \mathbb{N},\qquad u_{n+1}=\frac{e^{u_n}}{n+1} $$ converges to $0$ if $x \in ]-\infty,\delta[$ and diverges to $+...
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2 votes
0 answers
39 views

A Fundamental Theorem of Algebra type Result

Consider a polynomial $$p(t)=\sum_{i=1}^n a_it^{b_i}+a_0$$ of degree $b_n$, show that it admits at most $n+1$ nonnegative roots and at least one complex root. Assume all $b_i$ are positive and $a_i$ ...
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  • 4,393
0 votes
1 answer
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If $\Delta O \pmod {2^n} = (2)$ in $\Bbb{Z}/2^n$ for all $n \geq 1$, then does $\Delta O = (2)$ in $\Bbb{Z}$?

Let $O = $ the set of odd primes in $\Bbb{N}$. And $M = \Delta O = \{ x-y: x,y \in O\}$. Then we can take either set $X$ modulo $n$ for any $n \geq 2$: $\overline{X} = \{ x + (n) : x \in X\}$. Then ...
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1 vote
2 answers
224 views

About mapping class group.

What is best book for self learning mapping class group? I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit. Is there a topological space $X$ where we don't know $\...
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  • 4,081
1 vote
1 answer
193 views

Number of unordered factorizations of a non-square-free positive integer

I recently discovered that the number of multiplicative partitions of some integer $n$ with $i$ prime factors is given by the Bell number $B_i$, provided that $n$ is a square-free integer. So, is ...
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  • 393
1 vote
2 answers
99 views

What are the currently conjectures around symmetric group? [closed]

I am asking you this question: What are the currently conjectures around symmetric group on research? Indeed I am interested to work on unsolved problems concerning symmetric or alternating groups. ...
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0 votes
0 answers
65 views

Is it an open problem? (Sums of squares) [duplicate]

Has it been proven that if n can be represented as a sum of squares of two rational numbers, then it can also be represented as a sum of squares of two integers? This is a relatively simple statement, ...
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5 votes
1 answer
307 views

What is the true status of the Lehmer totient problem?

The Lehmer-totient problem : For a prime number $\ n\ $ we have $\ \varphi(n)=n-1\ $. In particular, we have $\ \varphi(n) \mid n-1\ $. Is there a composite number $\ n\ $ with $\ \varphi(n)\mid n-1\ ...
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1 vote
1 answer
51 views

Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}$

Question: Are there trigonometric and hyperbolic identities that are true in $\mathbb{R}$ but not true in $\mathbb{C}?$ For instance, $\cos^2(z)+\sin^2(z)=1$ is still true when we move to complex ...
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0 votes
1 answer
119 views

Current Open Problems Similar to the Basel Problem? [closed]

Are there any current open problems that are similar to the Basel problem?
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1 vote
0 answers
63 views

Searching for unsolved problems in the field of stability

I have proposed an approach for constructing Lyapunov functions for autonomous systems in my Ph.D. thesis and find some useful examples. Now, I am searching for some another example in this field or ...
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2 votes
0 answers
71 views

Reference Request : The number of binary string with some special condition (open problem)

(Sorry for my poor english...) I wonder the reference of Simon Marais Mathematics competition 2019 problem B4. This problem is as follow. (They said this problem is open problem.) B4. A set $\...
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  • 183
8 votes
0 answers
108 views

What is the current status of the snark theorem?

Wikipedia writes "W. T. Tutte conjectured that every snark has the Petersen graph as a minor... In 1999, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas announced a proof of this ...
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  • 338
0 votes
1 answer
110 views

Closed Nowhere dense is frentier of an open set

Excuse me can you see this question Every closed nowhere dense set is the frontier of an open set ... I tried on it but i am not sure , I prove it as follows Let $A$ is closed nowhere dense set, ...
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  • 85
0 votes
1 answer
211 views

Interpretation and Implications of the Abundance Conjecture

So I'm interested in working on the Abundance conjecture, which states that for every projective variety $X$ with Kawamata log terminal singularities over a field $k$, if the canonical bundle $K_{X}$ ...
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1 vote
1 answer
100 views

Open Problems to do with Polynomials and/or Elementary Function Theory

I was wondering what are some open problems (even if deemed impossible) in basic function theory (stuff you'd learn in high school) and/or open problems to do with polynomials... Thank you in ...
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