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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Conjecture: prime sum of two squares between every pair of consecutive squares

It appears that between every $n^2$ and $(n+1)^2$, for $n \geq 1$, there's at least one prime that is a Pythagorean prime and can be represented as the sum of two squares. In fact, it turns out that ...
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Is there any mathematical conjecture that is successfully applied in the real world in spite of being yet unproven?

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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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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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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73 views

Conjecture on primes

I am introducing you to a family of interesting conjectures. Before we begin, some notations. A $CRS_p$ is a complete residue set $mod$ $p$ (where $p$ is a prime), i.e. a set $M$, $M\subset\mathbb{N}$ ...
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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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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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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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Conjecture on the distribution of prime numbers and the prime counting function

I conjectured this result and i want to put it to the test. Let a perfect sequence be an arithmetic progression $(a_i)_{i=1}^{\infty}$ such that the sequence $(\pi(a_i))_{i=1}^{\infty}$ is also an ...
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27 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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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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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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101 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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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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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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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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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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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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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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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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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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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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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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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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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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Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?

The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions: An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated ...
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Expectation of arbitrary product of Brownian motions.

Let $(B_t)$ be a standard Brownian motion, $n\in\mathbb{N}$ even and $t_1,\ldots,t_n\in\mathbb{N}$. I was wondering whether there is a general formula for $$\mathbb{E} \left[ B(t_1)\cdots B(t_n)\right]...
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Questions about the Conjecture $ X Y Z $ [closed]

I really have a hard time asking this question. Because my mathematical background is almost at school level. I do not know in which theories of mathematics these questions are addressed. ...
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Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes

I wondered if it is possible to prove or refute the following conjectures involving finite products of Ramanujan primes that are denoted in this post as $R_j$, I add as reference the Wikipedia ...
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On a conjecture about the arithmetic function that counts the number of twin primes

In this post we denote (for a fixed positive integer or real number $x$) as $$\pi_2(x)=\#\{\text{ primes }p\leq x\,:\,p+2\text{ is also a prime}\}$$ the arithmetic function that counts the number of ...
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Conjecture $\frac{a}{a^r+b^r}+\frac{b}{b^r+c^r}+\frac{c}{c^r+a^r}\geq \frac{a}{a^r+c^r}+\frac{c}{c^r+b^r}+\frac{b}{b^r+a^r}$

following this kind of inequality One of my old inequality (very sharp) I propose this because I don't see it on the forum : Let $a,b,c>0$ and $a+b+c=1$ with $r\in(\frac{1}{2},1)$ and $a\...
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On conjectures about the arithmetic function that counts the number of Sophie Germain primes

In this post we denote (for a fixed positive integer or real number $x$) as $$\operatorname{Germain}(x)=\#\{\text{ primes }p\leq x\,|\,2p+1\text{ is also prime}\}$$ the arithmetic function that counts ...
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Pattern in Number of Conjugacy Classes of p-groups

I was playing around with the number of conjugacy classes of $p$-groups in GAP and made the following conjecture: If there is a group of order $p^{2n}$ with $k$ conjugacy classes then there is a ...
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Usefulness of the Hodge conjecture in mathematics and in physics?

If we assume that the Hodge conjecture is valid, what will it do for mathematics or for physics ? What consequences will this have in mathematics or in physics ? Thank you.
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For any $k \gt 3$, if $n!+k$ is a perfect power then does there exist any $n\gt k$?

A while ago, I asked a similar question For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true? where users mathworker21 and WE Tutorial School proved that for non-square $k$, $n\...
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Interesting property related to the sums of the remainders of integers

Cross-posted on Math Overflow too Let us define, $$r(b)=\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k})$$ After playing around with the $r(b)$ function for sometime I noticed that $r(b)$ ...
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For fixed hypotenuse, can the number of primitive Pythagorean triples exceed the number of non-primitive ones?

For the equation, $$a^2+b^2=c^2$$ if $c$ is fixed and the number of natural solutions for $a, b$ is greater than $1$, then can the number of primitive solutions (solutions in which $a, b, c$ are ...
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Can someone easily explain Artin's conjecture on primitive roots?

Wikipedia (1): In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. and ...
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Is this conjecture proven: there can be infinite arithmetic progression with $13$ terms and common difference $30030$, all its numbers primes?

I saw this conjecture in an old NT book by Sierpinski: There can be infinite arithmetic progression with 13 terms and common difference $30030 $ , all its numbers primes? It denotes that even one ...
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184 views

Interesting patterns related to the sums of the remainders of integers

Let us define, $$r(b)=\sum_{k=1}^{\lfloor \frac{b-1}{2} \rfloor} (b \bmod{k})$$ After reading the posts Surprising fact about a certain number-theoretic function and Do primes have special sums... I ...
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147 views

Weaker than abc conjecture invoking the inequality between the arithmetic and logarithmic means

In this post we denote the radical of an integer $n>1$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ with the definition $\operatorname{rad}(1)=1$. The abc conjecture is ...
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A Conjecture about the Density of an Additive Sequence of Integers mod $x$.

I have the following conjecture: Define $x$ mod $r$ as:$ \def\mod{\;\mathrm{mod}\;} \def\R{\Bbb{R}} \def\Q{\Bbb{Q}} \def\Z{\Bbb{Z}} \def\N{\Bbb{N}} $ $$\begin{align} \mod: \R\times\R^{\neq0} &\...

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