Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

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Reference request for conjecture about bridge and crossing number of knots

Murasugi in his book (Knot theory and its applications, page 60) writes: Conjecture. If $K$ is a knot, then $c(K) \ge 3(br(K) - 1)$, where equality only holds when $K$ is the trivial knot, the ...
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45 views

On the abundancy index of divisors of odd perfect numbers and a possible upper bound for the special/Euler prime

(Note: This post is an offshoot of this earlier question.) The topic of odd perfect numbers likely needs no introduction. Denote the sum of divisors of the positive integer $x$ by $\sigma(x)$, and ...
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49 views

Does $k=1$ follow from $I(5^k)+I(m^2) \leq \frac{43}{15}$, if $p^k m^2$ is an odd perfect number with special prime $p=5$?

The topic of odd perfect numbers likely needs no introduction. Denote the sum of divisors of the positive integer $x$ by $\sigma(x)$, and denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. Euler ...
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55 views

Conjecture about rational numbers

Inspired by normal numbers I created the simple following problem: First take a rational number, for example $\frac{3}{4}$ which is equal to $0.75$; now add the first digit after the decimal separator ...
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On Erdös-Szekeres convex polygons lower bound

I have problems with the construction of $2^{n-2}$ points that contain no n-gon, particulary, the proof of the book "Open Problems in Mathematics". The proof sais that: For $i = 0, ..., n-2$ ...
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If $N = q^k n^2$ is an odd perfect number with special prime $q$, then can $N$ be of the form $q^k \cdot (\sigma(q^k)/2) \cdot {n}$?

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $28$ are perfect since $$\sigma(6) = 1 + 2 + 3 + 6 = ...
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Beal's conjecture: confused about Wikipedia's page

Beal's conjecture states: If $A^x + B^y = C^z$ for $A, B, C, x, y ,z$ are positive integers and $x,y,z>2$ then $A$ , $B$ and $C$ have a common prime factor. As I understand, if for example $x$ is ...
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Conjecture on prime generating sequence [duplicate]

Statement Denote $p_n$ the $n$th prime. Consider the set $$\mathcal{P}=\{\sum_{i=1}^np_i:n\in\mathbb{N}\}$$ Prove that $|\mathcal{P}\cap\mathbb{P}|=\infty$ Why am I studying this First of all, I ...
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Problem from Discrete Mathematics and its application for Rosen section 4.4

This exercise outlines a proof of Fermat’s little theorem. a) Suppose that a is not divisible by the prime p. Show that no two of the integers 1 · a, 2 · a, . . . , (p − 1)a are congruent modulo p. b) ...
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Conjecture on prime gaps [duplicate]

I am presenting a conjecture i really like. Statement Consider $p_n$ the $n$th prime. Find wether there exist any $n$ such that $p_n-p_{n-1}=n$ and if there are any, fond wether there are infinitely ...
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NT conjecture 2' (on the sum of prime factors)

I made an article in which I present some conjectures in Number Theory (and Combinatorial Number Theory). I will post all the problems one by one, so you can express your opinion, give solutions, ...
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54 views

NT conjecture 2 (on the sum of prime factors)

I made an article in which I present some conjectures in Number Theory (and Combinatorial Number Theory). I will post all the problems one by one, so you can express your opinion, give solutions, ...
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1answer
121 views

NT conjecture 1 (on $\pi(n)$ and other functions)

I made an article in which I present some conjectures in Number Theory (and Combinatorial Number Theory). I will post all the problems one by one, so you can express your opinion, give solutions, ...
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53 views

Little conjecture in elementary number theory

I was calcultaing quadratic residues. I noticed nice pattern: quadratic residues modulo $p^n$ where $p$ is prime are well-behaved; they always repeat after $\frac{p^n - 1}{2}$-th place (starting with $...
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128 views

A conjectured criterion for a generalized ramanujan tau function

Given Ramanujan tau function $\tau(n)$, which is the nth Fourier coefficient of the modular discriminant $\Delta(q)=q\prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty \tau(n)\,q^n\tag{1a}$ Nothing ...
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Conjectured upper bound $\frac{\Gamma(x+n)}{\Gamma(x)}$ and limit at infinity

Let $x\geq 1$ a real number and $n\geq 3$ a natural number then we have : $$\frac{\Gamma(x+n)}{\Gamma(x)}\leq \Bigg(\frac{nx+\Big(\frac{n(n-1)}{2}\Big)}{\frac{n}{2}}-\Bigg(x^x(x+1)^{x+1}(x+2)^{x+2}...
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Is there an open mathematical conjecture which has been shown not to be provable in Peano Arithmetic before (possibly) being proved true?

There are several mathematical statements $\varphi$ about the natural numbers which are known to be true and known not to be provable in Peano Arithmetic (c.f. Gödel's incompleteness theorem, Paris-...
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A conjectured upper bound for $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n$ and $x\geq 1$

Hi I have (related https://mathoverflow.net/questions/337457/prove-that-left-fracxn1xn-11-rightn-left-fracx12-rightn ): Let $x\geq 1$ a real number and $n\geq 2$ a natural number then we have :...
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To prove the existence of solution(s) of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4$

To prove if a certain equation has a solution, we do not need necessarily solve that equation. Example: Is there any point of the curve $y=\sin^2(x)/x$ between $x=0$ and $x=\pi$ so that the slope of ...
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Is there any mathematical conjecture that is successfully applied in the real world in spite of being yet unproven? [closed]

I'm a philosophy student and I'm writing a thesis that makes a few comparisons between ethics and mathematics. My knowledge of mathematics is limited, however, and in the process of making my ...
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Prove a conjecture, balls in boxes, n steps

My uncle gave me the following problem to work on (just for fun), he doesn't know whether the problem has a solution. I haven't been able to solve it and I give up, I don't think my current knowledge ...
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$\sum_{i=1}^n ( nGx_i^{G} + G^{x_i}) \ge n^2G + G^2n$, for all $x_i>0$, where $G=\prod_{j=1}^nx_j$. [closed]

Prove or disprove that, for all $x_1,x_2,\ldots,x_n>0$, it holds that $$\sum_{i=1}^n ( nGx_i^{G} + G^{x_i}) \ge n^2G + G^2n, \space \space \space \text{where} \space \space \space G=\prod_{j=1}^...
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Grimm's Weak Conjecture

On Wikipedia it says that the following weaker version of Grimm's conjecture is also still open: "A product of $k$ consecutive composite numbers has $k$ pairwise distinct prime factors." Is it true, ...
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27 views

Question related to Sophie Germain Primes

Assuming $p_i$ is a prime consider the sequence defined by $q_1=p_i$ and $q_n=2\ q_{n-1}+1$ for $n>1$. Let $d$ be the greatest value $n$ for which $q_1..q_n$ are all Sophie Germain primes which I ...
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Conjecture about Jensen's inequality and polynomials

Hi it's related to the following conjecture An inequality for polynomials with positives coefficients : We have the first conjecture : Let $x,y>0$ then we have : $$(x+y)f\Big(\frac{x^2+...
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68 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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157 views

How do we prove this continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following continued fraction holds $$\frac{\displaystyle4\Gamma\left(\...
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53 views

Conjecture: There are infinitely many $N \in \Bbb{N}$ such that $p$ a prime $p \leq \sqrt{N+1} \implies p \mid N$?

Conjecture. There are infinitely many $N$ such that if $p$ is a prime $\leq \sqrt{N+1}$ then $p \mid N$. Is this another hard to prove number theory conjecture, or do you have some idea of how to ...
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29 views

A pattern with third coefficients of sums of powers.

So I have heard countless times of Bernoulli Numbers and its relation to the sums of powers. Inspired by this (as well as Chebyshev polynomials), I decided to look at the sums of powers myself and ...
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38 views

Is the following an existing conjecture or a conjecture at all?

the following floated to my mind today, can you verify if it stands to be true, or is a pre-existing conjecture. If not, can you correct me? And if it is one, can you prove it? Prime factorisation of ...
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Conjecture: every set of $n$ equally spaced integers $\leq n^2$ contains a prime.

My claim: Any set of exactly $n$ integers in the range $[1,n^2]$ which are an arithmetic sequence and share no common factor will contain at least one prime. Let $n\geq 2$ be a natural number, and ...
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42 views

Examples of false conjectures suggested by probabilistic evidence

Background Recently I read about some attempts of finding the exact order of Mertens function. It is conjectured that $$0<\limsup_{x\to\infty}\frac{M(x)}{\sqrt x(\log\log\log x)^{5/4}}<\infty\ (...
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Conjectured series representation of elliptic integral powers $K(k)^n$

This is a very simple question which has probably already been answered; I apologize if that is the case. We know that the following equation holds: $$K(k)=\frac{\pi}{2}\sum\limits_{n=0}^\infty a_n^{...
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Bass-Quillen conjecture for non-affine case

Bass-Quillen conjecture expects that any vector bundle on $U\times \mathbb{A}^1$, to be extended from $U$. Here $U$ is a regular affine scheme. Being affine is an essential part of the conjecture, you ...
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1answer
112 views

Evaluate $S_n=\frac{1}{\log(2)}+\frac{2}{\log(3)}+\frac{3}{\log(4)}+\frac{4}{\log(5)}+\cdots+\frac{n}{\log(n+1)}$

Hi I want to evaluate the following sum : $$S_n=\frac{1}{\log(2)}+\frac{2}{\log(3)}+\frac{3}{\log(4)}+\frac{4}{\log(5)}+\cdots+\frac{n}{\log(n+1)}=?$$ My try : Using a well know trick we have : $$\...
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176 views

A property implying union-closed sets conjecture condition

Given a union-closed family $\mathcal{F}$ of $n=\vert \mathcal{F} \vert$ sets, $n$ odd, and its family $\mathit{J}(\mathcal{F})$ of $m = \vert\mathit{J}(\mathcal{F})\vert$ basis sets (or $\cup$-...
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Expressing continued fractions through $e$

The following are some conjectures of mine that I have discovered empirically. The last three conjectures are true if the first four are true, and vice versa. i. $$e=3-\cfrac{1}{4-\cfrac{2}{5-\...
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Defined intervals conjectured to contain at least one pair of twin primes

Apologies up front for a lengthy post. The questions posed are simple, but the thoughts underpinning them require careful exposition. Definitions: The twin prime pair $(3,5)$, not being of the form $...
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An inequality that involves consecutive primes, prime gaps and roots of prime numbers as a weak form of Firoozbakht's conjecture

In this post for integers $n\geq 1$ we denote the $n$-th prime number as $p_n$. When we consider that $k>1$ runs over integers, from the theory of the Stolarsky mean we can deduce that as $k\to \...
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88 views

If $C_2$ is irrational, then there are infinitely many twin primes?

This is a natural follow-up after question 3629282. It is trivial that the irrationality of Brun's constant $B_2\approx1.90216$ implies that there are infinitely many twin primes: $$ B_2 \mbox{ is ...
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1answer
61 views

Conjecture on a functional equality and nested radical of Ramanujan :$3=\sqrt{1+f(2)\sqrt{1+f(3)\sqrt{1+\cdots}}}$

Hi i was wondering something about the great Ramanujan : I think moreover I'm not the only one who propose this kind of problem (so if you have a link related to this subject). We have : $$3=\sqrt{1+...
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1answer
42 views

On proving an infinite-nested radical

I was playing around with square roots and I noticed that the number $1$ can be seemingly expressed as an infinite nested radical with an easy pattern. I then noticed that if this is true, this would ...
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216 views

Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Recall that the Thue–Morse sequence$^{[1]}$$\!^{[2]}$$\!^{[3]}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its ...
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43 views

list of open conjectures which are solved by students in high-Middle school level or receprocally?

I know only Snevily's conjecture which it was proved in 2009 by Bodan Arsovski. and He was a high-school student at the time., Are there other conjectures which are solved by students in high ...
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45 views

If $q^k n^2$ is an odd perfect number with special prime $q$, then $k \neq 1$ (MSE version)

Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
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1answer
135 views

How were the five large solutions to the Fermat-Catalan conjecture found?

The Fermat-Catalan conjecture is the statement that the equation$a^m + b^n = c^k$has only finitely many solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers ...
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30 views

Attempt to get a characterization of even perfect numbers from an equation involving the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(n)$ for integers $n\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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191 views

Conjecture Prove that : $\sum_{cyc}\frac{a}{a^n+1}\leq \sum_{cyc}\frac{a}{a^2+1}\leq \frac{3}{2}$

Conjecture, Prove that : $$\sum_{cyc}\frac{a}{a^n+1}\leq \sum_{cyc}\frac{a}{a^2+1}\leq \frac{3}{2}$$ Under the assumptions $a\geq b\geq 1\geq c>0$ such that $abc=1$ and $\frac{c}{c^n+1}\...
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933 views

Conjectured continued fraction formula for Catalan's constant

Yesterday I posted this conjecture, but then deleted it thinking it was false. Turns out Python doesn't define $a^b$ as a^b, but rather as ...
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2answers
61 views

Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$?

Is the last digit of $2^{2^{n-1}(2^n-1)}-1$ always $5$ for $n >3$? I did modification to the Mersenne numbers (Even perfect numbers) foruma I put that formual to be the power of 2 have got : $2^{...

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