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Questions tagged [conjectures]

Use this tag if your question is about a well-known conjecture or a conjecture of your own.

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Conjecture about testing if $A$ is a UFD.

Consider an integral domain $A$ with a norm $N$. For all elements in $A$. Since we are talking about an integral domain there are no zero-divisors. Thus $N(x) = 0 $ iff $x=0$. Let $k$ be the ...
2
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2answers
52 views

Is there a counter-example to these number theoretic conjectures?

Question and Summary I recently made the following heuristic observations: Let, $$ xy = p_1^{a_1} p_2^{a_2} \dots p_n^{a_n} $$ where $a_i\geq1$ Conjecture $1$: then there must exist $x-y=p_{n+1}$ ...
2
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1answer
97 views

A conjecture about irreducible polynomials with integer coefficients

Let $f\in\mathbb Z[X]$, define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments. Theorem: If $f\in\mathbb Z[X]$ is non constant and reducible ...
0
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1answer
43 views

Hopf Conjecture about Curvature and Topology

Hopf Conjecture states that: If even-dimensional manifold $M$ admit a metric of positive (non-negative) curvature then its Euler characteristic is positive (non-negative). My question is about non-...
1
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0answers
69 views

Multiplicative order of $a\bmod c^{k+1}$

I have some questions about moving from $\mathbb Z_{c^k}$ into $\mathbb Z_{c^{k+1}}$-specifically, with regard to the order of elements. Suppose $a$ (which is coprime to $c$) has multiplicative ...
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344 views
+50

Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
1
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3answers
69 views

Proof of $ x^2 + y = y^2 + x$ when $ x+ y =1$ and x is larger than y

I know that whatever numbers you choose for x and y and their sum equals to 1 will satisfy the equation $x^2 + y = y^2 + x$ Algebraic proof: Given: $x + y = 1$ $$LS = x^2+ y = (1-y)^2 + y = 1 -...
2
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1answer
41 views

Conjecture about fixed prime divisors of polynomials with integer coefficients

While experimenting with random polynomials I've found this conjecture: A polynomial $f\in\mathbb Z[X]$ of degree $n$ with co-prime coefficients have no fixed prime divisor $p> n$. A fixed ...
1
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0answers
80 views

Variation Of The Collatz Conjecture Discovered [closed]

Consider the following operation on an arbitrary positive integer: If the number is divisible by 12, divide it by 12. If the number is divisible by 10, divide it by 10. If the number is divisible ...
2
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1answer
49 views

Conjecture about polynomials $f_n\in\mathbb Q[X_1,\dots,X_n]$ defining bijections $\mathbb N^n\to\mathbb N$

This is inspired by an answer of a question of mine: Bijective polynomials $f\in\mathbb Q[X_1,\dots,X_n]$ There is a polynomial $f_1\in\mathbb Q[X_1]$ which define a bijection $f_1:\mathbb N\to\...
5
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2answers
55 views

What is the sufficient and necessary conditions that $-1 \in G$, where $G$ is a multiplicative group of a ring.

I am trying to prove the following conjecture. Let $(R, +,\times)$ be a finite ring with an identity. Let $G$ be a subgroup of $(R,\times)$ with order $d$. Then $-1\in G$, if and only if $2\mid ...
1
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1answer
60 views

I have found a way of computing Euler's number. Is there any possible intuition of how that might be the case?

So a few days ago I just kind of messed around with my calculator, when I had an idea about a new continued fraction. I inputted it, and I found that it converged really quickly, and, quite wondrously,...
1
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1answer
52 views

Probability for composite $n$ to have prime factor $\geq \sqrt n$

Let $\operatorname{GPF}(n)$ denote the largest prime factor of $n\in\mathbb N_{>1}$. My computer tests for intervalls $[m,n]$, where $n<10,000,000$, suggests that the probability $\operatorname{...
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0answers
38 views

Inequality of matrix infty-norm: Does $\|(I - L)^{-1}U\|_{\infty} \leq \|H\|_{\infty} < 1?$

If $H\in \mathbb{R}^{n\times n}$ is a matrix with $\|H\|_{\infty} < 1$ and $H = L + U$, where $L$ is the strict lower triangular part of $H$ and $U$ is the upper triangular part of $H$, can we ...
1
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1answer
46 views

Is this variant of Goormaghtigh's conjecture known?

Goormaghtigh's conjecture states that the only non-trivial integer solutions of $$ {\frac {x^{m}-1}{x-1}}={\frac {y^{n}-1}{y-1}} $$ satisfying ${\displaystyle x>y>1}$ and ${\displaystyle n,m&...
4
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1answer
35 views

On an interesting assertion in the OeisWiki page on multiply-perfect numbers

The following (interesting) assertion appears in the OeisWiki page on multiply-perfect numbers: ...
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2answers
43 views

Every 'decreasing' or 'increasing' infinite sequence whose sum converges contains at least one term of magnitude 0

Note on the title: This conjecture is not restricted to real numbers, 'decreasing' and 'increasing' were used because of the character limit. The actual conjecture is that every sequence whose terms ...
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1answer
49 views

Does the infinite product of the reciprocals of a decreasing (or increasing) function equal zero? [closed]

I've made a short document explaining what I've just claimed. I'd like to know if the criteria for the infinite product to be zero is enough to hold for all decreasing and increasing functions. ...
3
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2answers
119 views

For all $n>1$ there are positive $a+b=n$ such that $a+ab+b\in\mathbb P$

For all integers $n>1$ there are positive integers $a,b$ such that $a+b=n$ and such that $a+ab+b\in\mathbb P$. Tested for all $n\leq 1,000,000$. Hopefully, someone can explore and explain the ...
0
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0answers
77 views

Where is the mistake: On the sum of two prime numbers.

Someone could help me find some error in the reasoning: We know, that the canonical decomposition of $n!$ is: $n!=\prod_{p_{i}\leq n}p_{i}^{\alpha_{i}(n)}$, where: $\alpha_{i}(n)=\sum_{t=1}^{r}[\...
2
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1answer
43 views

What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function?

The title says it all. What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function? I tried using Sage Cell Server to ...
0
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1answer
46 views

A question regarding a paper of Ochem and Rao about the radical of an odd perfect number

Let $\operatorname{rad}(n)$ denote the radical or square-free part of the positive integer $n$, that is, $$\operatorname{rad}(n) = \prod_{p \mid n}{p}$$ where $p$ runs over primes. In the paper ...
1
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1answer
58 views

The sum of all divisors of $N$ is a 2 potency iff $N$ is a product of different Mersenne primes

As far as I have controlled: $\sigma(a)=2^n$, for some $n\in\mathbb N \iff $ $a$ is a product of different Mersenne primes. The $\Leftarrow$-part is an immediate consequence of that $\sigma$ is ...
0
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0answers
57 views

Which numbers will iterate to others under the Collatz iteration?

I have a question about the Collatz conjecture and how some numbers merge trajectories. Take the standard map: $$C(n) = \begin{cases} n/2, & \text{if $n \equiv 0$ mod $2$} \\ 3n+1, & \text{...
1
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1answer
37 views

A weaker conjecture than a known conjecture

I really apologize if my question is not appropriate here, though I hope it is. Let $C$ be any known conjecture in mathematics, which is still open. Let $D$ be another conjecture such that a positive ...
0
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1answer
90 views

Where is the flaw in this proof of Legendre's Conjecture?

Introduction The following argument has been advanced by one of my friends which attempts to prove the Legendre's Conjecture. I could find no flaw in the argument and so I am posting it here in the ...
3
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0answers
32 views

$\sum_i X_i^2$ has $\chi^2_{n}$ distribution and $X_i$ i.i.d. imply $X_i$ normal

Let $X_1,\ldots,X_n$ be i.i.d. random variables with distribution $F$. It is known that if $F$ is the standard normal distribution then $$ S:=\sum_{i=1}^n X_i^2 $$ has a chi square distribution with $...
0
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1answer
54 views

When given a non-multiple of $3$, $k$, is it possible construct $m<k$ with these conditions? [closed]

This is another Collatz-related problem about trying to represent a number in a certain form. As is usually the case with the Collatz conjecture, this is probably not useful. My question is : Can ...
4
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1answer
87 views

If $x$ and $y$ are solitary numbers satisfying $\gcd(x,y)=1$, under what conditions does it follow that $xy$ is also solitary?

Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$. Denote the abundancy index of $z$ by $I(z) = \sigma(z)/z$. If the equation $I(z)=I(a)$ has the lone solution $z=a$, then $a$ is said ...
2
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2answers
180 views

Can any integer, not a multiple of three, be represented as $n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$?

This question has some relevance to the Collatz conjecture. It was originally based on trying to represent a number like this: Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+...
2
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3answers
54 views

Divisibility rule

Example: $2^1$=2 --> $2\mid2$ If a number has their last digit divisible by 2, than the number is divisible by 2 $2^2$=4--> $4\mid2$, $4\mid4$ If a number has their last two digit divisible by ...
1
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1answer
60 views

Can this bound for the abundancy index of $n$ be improved, given that $q^k n^2$ is an odd perfect number with $k=1$?

In what follows, set $I(x)=\sigma(x)/x$ to be the abundancy index of $x \in \mathbb{N}$, where $\sigma(x)$ is the sum of divisors of $x$. If $I(y)=2$ and $y$ is odd, then $y$ is called an odd perfect ...
3
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0answers
63 views

Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
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0answers
77 views

Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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2answers
49 views

Unique factorization conjecture?

Let $A_p$ be an Integral domain. Conjecture : If every $a$ in $A_p$ that equals $b \space c$ for irreducible elements $b,c$ in $A_p$ , has Unique factorization then the Integral domain $A_p$ is a ...
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0answers
72 views

Fortune's conjecture solved for limited cases?

I am not a mathematician, but while doing other work, I came across the Fortune conjecture. According to Wikipedia and other research, it seems that it has not yet been proven. I thought about the ...
2
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0answers
30 views

Schwarz inequality for unital positive maps on C*-algebras

I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that Let $\phi : \mathcal{A} \to \mathcal{B}$ be a unital positive linear ...
6
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1answer
71 views

Validation for a conjecture about Chinese Remainder Theorem for groups

I was wondering if the following statement is true: Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $i\neq j$. Then $G/H_1\cap H_2...\cap H_n\cong G/H_1 \times......
0
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2answers
57 views

Conjecture for $\ln(x)$ and $\frac{d^ny}{dx^n}$

I'm not really sure where to start, I found the first, second, third and fourth derivatives of $\ln(x)$ to be $\frac{1}{x}$,-$\frac{1}{x^2}$, $\frac{2}{x^3}$, and -$\frac{6}{x^4}$, respectively. ...
6
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1answer
102 views

Calculating the $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}$

Question: If $s \in \mathbb{N}$ is it true: $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}={\zeta\left[{s+1 \choose 2}\right] \above 1.5pt \prod_{k=1}^s{s \choose k}};$ where $\...
5
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2answers
68 views

Primes of the form $p_1p_2\dotsm p_k+1$ [closed]

Let $p_1,p_2,...,p_k$ be the first $k$ primes and $k>1$. Does there exist a prime $p_{k+1}$ such that $p_{k+1} = p_1p_2...p_k + 1$?
3
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1answer
101 views

What is the common (and simplified) value for $D(q^k)D(n^2) = 2s(q^k)s(n^2)$ when $q^k n^2$ is an odd perfect number?

In an answer to an earlier question, it is shown that $$D(2^p - 1)D(2^{p-1}) = 2s(2^p - 1)s(2^{p-1}) = 2^p - 2,$$ if $2^{p-1}(2^p - 1)$ is an even perfect number, $D(x) = 2x - \sigma(x)$ is the ...
2
votes
1answer
44 views

Manufacturing desired planks from an existing pile of planks

Suppose there is a pile of commensurable planks that only may differ in lengths $0<a_1\leq\cdots\leq a_m$, which are to be used to manufacture planks of length: $0<b_1\leq\cdots\leq b_n$. ...
11
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1answer
172 views

Is there are similar conjecture like this??

Talking with my friend, my friend suggest impressive conjecture that For $i\in\mathbb{N}$, there are always exist natural number $r$ that satisfies $$\sum_{n=1}^{i} \frac{1}{n^r}=\frac{p}{q} , \...
3
votes
1answer
39 views

Free action of finite cyclic group on $S^3$

I'm starting to read Allen Hatcher's Space of Knots and at the end of the first paragraph he says that his work would apply to all knots if the following conjecture is true: Every free action of a ...
3
votes
1answer
159 views

A conjecture on the closeness of twin primes

Let $p_1$ and $p_2$ be twin primes, and let $p_1-1=a_1\times b_1$ and $p_2+1=a_2\times b_2$ be such that $|b_1-a_1|$ and $|b_2-a_2|$ are minimised. Similarly, let $p_1+1=p_2-1=a\times b$ be such that ...
0
votes
1answer
109 views

Subsubgroups are subgroups of subgroups / Multiplicative Property of the Index

Algebra by Michael Artin Prop 2.8.14 Multiplicative Property of the Index Statement of Prop 2.8.14 Let $G \supseteq H \supseteq K$ be subgroups of a group G. Then $[G:K] = [G:H][H:K]$. Proof of ...
2
votes
0answers
67 views

Inquiry about the Wolfram MathWorld entry on Odd Perfect Number

I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number. Last sentence of the fifth paragraph states that: Hagis (1980) showed that odd perfect numbers must have at ...
1
vote
1answer
69 views

There are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$

Conjecture: Given $a,b\in\mathbb Z^+$ there are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$. I would like help with a proof or a counter-example. Tested for millions of pseudo random numbers.
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0answers
84 views

A problem inspired by the Twin Prime conjecture

I came up with a question several hours ago...but I couldn't find any information about it. The problem goes like below $$P^n_k =\{(p_1,...,p_n)|p_1<...<p_n:primes,p_n-p_1\le k\}$$ $$k_n=min\{...