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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4
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2answers
163 views

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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28 views

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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1answer
71 views

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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23 views

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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0answers
129 views

On a conjecture about the arithmetic function that counts the number of twin primes

I've asked the same question on MathOverflow two days ago as On a conjecture about the arithmetic function that counts the number of twin primes, I add this reference while I hope to know what about ...
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1answer
61 views

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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56 views

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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88 views

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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71 views

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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259 views

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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2answers
75 views

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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1answer
40 views

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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37 views

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 ...
3
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2answers
185 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 ...
3
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1answer
149 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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18 views

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} &\to ...
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1answer
132 views

Enrique Santos L's “Proof that no odd perfect number exists”

Background Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$. Let $n$ be an odd perfect number given in the so-called Eulerian ...
6
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1answer
610 views

Improvement on the concept of separating families for the union-closed sets conjecture?

The union-closed sets conjecture states the following. Let $F$ be a finite family of finite sets that is union-closed and let $\cup (F)$ be the union of all sets in $F$. Then we can find an element in ...
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1answer
48 views

Possible relationship between non-divisors of odd perfect numbers and coefficients of corresponding cyclotomic polynomials?

A positive integer $n$ is called perfect if $\sigma(n)=2n$, where $$\sigma(n)=\sum_{d \mid n}{d}$$ is the sum of divisors of $n$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect ...
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On expressing $\frac{\pi^n}{4\cdot 3^{n-1}}$ as a continued fraction.

It is a celebrated equation that $$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\ddots}}}}$$ However, there are two other conjectured equations that I found which, if true (...
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1answer
135 views

On conjectured continued fractions and $e$

Playing around with numbers, I conjectured three incredibly interesting things: $$9+\cfrac{1}{18+0\times 12\cfrac{1}{18+1\times 12+\cfrac{1}{18+2\times 12+\cfrac{1}{18+3\times 12+\ddots}}}}=\frac{4e^{...
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1answer
44 views

Prove that $\sum_{k=2}^{n}\{\sqrt{a_{k-1}}+\sqrt{a_{k}}\}^{-1}=\frac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}$

If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ are consecutive terms of an arithmetic progression (None of which are $0$) , prove that $\sum_{k=2}^{n}\{\sqrt{a_{k-1}}+\sqrt{a_{k}}\}^{-1}=\frac{n-1}{\sqrt{a_{1}...
4
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1answer
158 views

Formula for Feigenbaum’s constant?

I have conjectured a formula to calculate the Feigenbaum constant $\delta \approx 4.66920$. $\delta\stackrel{?}{=}$ $$4+\cfrac{1\times 2 -1}{1+\cfrac{2\times 3 -1}{2^2+\cfrac{3\times 4 -1}{1+\cfrac{...
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93 views

Minimal counterexample to $\frac{p - 1}{p^2}$-conjecture

There existed once a “folklore” conjecture that stated: Suppose $p$ is a prime. Then any finite group $G$ with $> (1 - \frac{p-1}{p^2})|G|$ elements of order $p$ has exponent $p$ This ...
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165 views

Is this elementary-number-theoretic conjecture already mentioned somewhere or it is entirely new?

While I was doing some research in elementary number theory I discovered some regularities that seem to be very promising. First, function $d$ can be defined as $d(n)=d(\prod_{i=1}^{k(n)} p_i^{r_i})=...
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2answers
137 views

Conjectured summation inequality

I was playing around with numbers and noticed that the following series is quite close to $\sqrt 2$... but not quite. So I have conjectured that $\sqrt 2$ is arguably the closest highest bound; $$1+\...
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38 views

Decimal repunits of order greater than 2 do not have repunits in any other base

This is a proof of a special case of [Goormaghtigh's conjecture][1],namely $$\frac{10^m-1}{10-1} = \frac{y^n-1}{y-1}$$ satisfying $10>y>1$ and $ n,m >2 ...
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1answer
114 views

$am+bn$ is prime only if $\gcd(m+n,a-b)=1$

I've discovered some conjectures of the form All big enough odd integers $k$ has partitions $k=m+n$ such that $f(m,n)$ is prime, for different functions $f:\mathbb N\times\mathbb N\to\mathbb N$...
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0answers
182 views

Conjecture: Any odd integer $k>1$ has a partition $m+n,\,m,n>0,$ such that $m^2+n$ > is a prime. [closed]

I want some thoughts about this conjecture: Any odd integer $k>1$ has a partition $m+n,\,m,n>0,$ such that $m^2+n$ is a prime. Tested for $k<100,000,000$. There seems to be a ...
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1answer
32 views

Is there an error in this attempt at a counter-example to the thrackle conjecture?

I spent some time today thinking about Conway's thrackle conjecture. After a while, I came up with this graph, which looks to me like a counter-example. I'd assume there's an error in there that I've ...
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1answer
38 views

What does Chowla's cosine problem actually ask to find?

The problem according to this description is: Let $ A \subseteq {\mathbb N} $ be a set of $ n $ positive integers and set $m(A) = - \min_x \sum_{a \in A} \cos(ax).$ What is $ m(n) = \min_A m(A) $? ...
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1answer
134 views

Conjecture about a continued fraction

Conjecture: $$\large 2^{n-1}+\frac{1}{2+\cfrac{1}{2^{n}-1+\cfrac{1}{2+\cfrac{1}{2^{n}-1+\cfrac{1}{2+\ddots}}}}}=\frac{1+\sqrt{3a_n}}{2}\tag*{[1]}$$ such that $a_n=4a_{n-1}+1$ and $a_0=0$.$\quad(n\...
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0answers
112 views

How do I solve the Continued Fraction $[1,x,x^2,x^3,\ldots]$

My function $M(x)$ is defined as $$M(x)=1+\cfrac{x^0}{1+\cfrac{x^1}{1+\cfrac{x^2}{1+\cfrac{x^3}{1+\cfrac{x^4}{1+\cfrac{x^5}{1+\cfrac{x^{6}}{1+\cfrac{x^{7}}{1+\cfrac{x^{8}}{1+\cfrac{x^9}{1+\cfrac{x^{10}...
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1answer
271 views

Conjecture : an odd perfect square $n>1$ raised to the third power is never divisible by the sum of $n$'s divisors

This is a conjucture that I created : Let $\,\,n = (2k+1)^2 \,\, $with $k\in \mathbb{N}$ and so $n>1$, and let $$\,\,A = \sum_{d \in \mathbb{N}; \ d|n} d.$$ Then $n^3$ is never divisible by ...
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1answer
287 views

$\sum_{i=1}^n\frac{x_i}{\sqrt[nr]{x_i^{nr}+(n^{nr}-1)\prod \limits_{j=1}^nx^r_j}} \ge 1$, for all $x_i>0$ and $r \geq \frac{1}{n}.$

Prove that, for all $x_1,x_2,\ldots,x_n>0$ and $r \geq \frac{1}{n}$, it holds that $$\sum_{i=1}^n\frac{x_i}{\sqrt[nr]{x_i^{nr}+(n^{nr}-1)\prod \limits_{j=1}^nx^r_j}} \ge 1.$$ This is a slightly ...
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1answer
54 views

A Weak Schönflies Theorem in $\mathbb{R^n}$?

I was reading about the classical topological result of Schönflies that Jordan curve in a plane can be extended homeomorphically onto the whole plane. The well-known counterexample of Alexander's ...
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36 views

Erdős-Strauss Conjecture: Is there a simpler way to solve? [duplicate]

I'm a huge fan of conjectures, and I'm fascinated by this new one. The Erdős-Strauss conjecture is that $\frac{4}{n} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$, where $n$ is greater than or equal to $...
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0answers
143 views

proof of prime in every interval $(p^2,p^2+p)$

Overview We'll introduce a sort of little hack called the missing modulo conjecture which can identify an integer's previous prime. We then show that this may not be perfectly reliable on account of ...
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1answer
74 views

The Rook Conjecture: arrangement of $p$ primes being distinct $\pmod{p}$ through $p^2$

For any prime $p$, divide $[1,p^2]$ into $p$ equal intervals of length $p$, so that the first interval is $[1,p]$, the next $[p+1,2p]$, and so on. It is definitely unproven but seems likely that there ...
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1answer
48 views

A conjecture concerning Pollard rho

I'm investigating the Pollard rho factorization algorithm, searching for a proof that any odd composite can be factorized for some start number. I'm only studying the standard function $f(x)=x^2+1\...
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1answer
62 views

A conjecture in number theory with twin primes

It's a conjecture found with the help of Wolfram Alpha : Let $p_i$ be the first primes and $n> 5$ with $n$ an odd natural number numbers we are interested by the quantity : $$A=(1+p_1\times ...
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1answer
130 views

Do all elements of $[n+1,2n]$ have strictly higher gpf than elements of $[1,n]$ when sorted by gpf?

For any $n\in\mathbb N$, let $A=\{x\in\mathbb N \mid 1 \leq x \leq n\}$ and $B=\{x\in\mathbb N \mid n+1 \leq x \leq 2n\}$. Order the sets by greatest prime factor of each element, ascending. Let $...
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1answer
166 views

Prove or Disprove the Conjecture on Cousin Primes

Here is the problem in a nutshell - Ethers or Ether of n, $\mathbf{\eth(n)}$: Ethers of any number n is the set of all the values of every possible permutations of its digits. Illustration Ethers of $...
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1answer
156 views

Greatest common divisor of consecutive square free numbers

I guess that every prime number occurs as the greatest common divisor of two consecutive square free numbers, which I don't expect a proof of. But I've done some experiments indicating that: If $m,...
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0answers
62 views

A conjecture about a smothness concerning consecutive primes

This is a conjecture inspired by the relative but piecewise smoothness of the sequence https://oeis.org/A068901/graph?png=1, about the least number $a_n$, such that $n\mid p_n+a_n$, where $p_n$ is the ...
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2answers
510 views

For any $k \gt 1$, if $n!+k$ is a square then will $n \le k$ always be true?

In Dabrowski's paper, he showed that it would follow from the abc conjecture that the equation $$n!+k=m^2$$ has a finite number of solutions $n, m$ for any given $k$ which was my motivation to find ...
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0answers
75 views

Can gcd separate primes from composite numbers?

Let $x_i=i+1, 1\le i \le n$, $y_i = +1,$ if $x_i$ is prime, $-1$ otherwise. Conjecture: Then (?) there exists $b,c_i \in \mathbb{R}, 1 \le i \le n$ such that for all $j=1,\ldots,n$: $$y_j = \text{ ...
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0answers
49 views

What are the examples of results which have been proved assuming a conjecture to be both true and false?

Littlewood proving that $Li(x) - \pi(x)$ changes sign infinitely often by assuming the Riemann Hypothesis. But later he was able to prove it assuming the Riemann Hypothesis is false. I find it ...
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1answer
75 views

Do all the zeroes of this arithmetic function occur at even integers?

Let $a(n)$ be the number of natural numbers $\le n$ which have an odd number of distinct prime factors. Let $b(n)$ be the number of natural numbers $\le n$ which have an even number of distinct prime ...

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