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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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A new way to look at the Collatz conjecture

First of all, I this isn't a proof but rather a new way to look at the Collatz (which hopefully other mathematicians can make a proof/counterproof out of). What I came here for is to know if what I ...
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20 views

Circumference of convex curves in the plane

For triangles, rectangles and ellipses in the plane the quote between the circumference and the diameter is invariant when the figures are magnified. The diameter is the maximum distance between ...
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1answer
79 views

Group (mathematics) Conjecture

Given $(G,•)$ as a Group with finite set $G$, operator •. Define: subset $S \subset G$ is called the core of the Group if and only if $$ \{ x•y ~|~ x \in S, y \in S \} = G \setminus S$$ ...
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1answer
54 views

A few questions regarding the function $f(x) = x+\exp(x)\cdot \log(x)$

The function $f(x) = x+\exp(x)\log(x)$ occurs prominently at Lagarias inequality: $\sigma(n) \le H_n + \exp(H_n)\log(H_n)$ where $\sigma(n)$ is the sum of divisors, and $H_n$ is the n-th harmonic ...
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2answers
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Breaking the barriers at $q=5$ and $q=13$ for $q^k n^2$ an odd perfect number with special prime $q$

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. If $\sigma(N)=2N$ and $N$ is odd, then $N$ is called an odd perfect number. The question of existence of odd perfect numbers is the ...
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130 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
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45 views

Equivalence of Frankl's Conjecture

Frankl’s conjecture is one of the most famous problems in combinatorics. Frankl's conjecture claims: For every finite non-empty set $A$ and for every Frankl's family $F$ on $A$ exists $a\in A$ such ...
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1answer
66 views

Bounds for the $n$-th record gap between primes in a residue class

(Following question 2269073.) Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ r+3q, \ldots \tag{P} $$ Dirichlet proved that there ...
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1answer
87 views

Which other unsolved problems, have necessary restrictions on the prime gaps?

We all know of Unsolved problems, like Goldbach,Legendre, and Grimm's conjectures. Goldbach has the necessary condition of: There exists a prime between $n$ and $2n-2$, which means prime gaps are ...
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1answer
129 views

New property of Mersenne primes?

While playing with Mersenne numbers, I found the following property distinguishing Mersenne prime numbers from Mersenne composite numbers. A Mersenne number, $\text{M}p$, is a number of the form $2^p ...
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1answer
2k views

Could prime numbers be defined like this?

While playing with prime numbers, I found the following definition. Let $p$ be an integer. Then $p$ is a prime number if and only if there is some integer $b \neq 1$ such that $$ \frac{b^p - 1}{b - 1} ...
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2answers
362 views

Sum of digits of $a^b$ equals $ab$

The following conjecture is one I have made today with the aid of computer software. Conjecture: Let $s(\cdot)$ denote the sum of the digits of $\cdot$ in base $10$. Then the only integer ...
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23 views

A consequence of assuming the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

Let $x$ be a positive integer. Denote the sum of divisors of $x$ by $$\sigma(x) = \sum_{d \mid x}{d},$$ and the deficiency of $x$ by $$D(x) = 2x - \sigma(x).$$ A number $N$ is said to be perfect if $...
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32 views

Conjectured “weighted resolution of identity”?

Background I was recently wondering if there was an extension to this formula I found after reading Abdelmalek Abdesselam's answer. I think I managed find an equivalent of it. Question We know ...
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0answers
46 views

Does the Graceful tree conjecture refer to all or only some trees?

I am a total beginner in this field and i‘m not really versed in the terminology, so please bare with me. What I know, is that a graceful labeling, refers to a tree with $n$ vertices, where each ...
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3answers
88 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
105 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 ...
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1answer
54 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-...
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0answers
79 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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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 $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
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3answers
76 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 -...
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1answer
49 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 ...
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91 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
51 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\...
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2answers
57 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 ...
3
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2answers
187 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,...
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1answer
57 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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1answer
51 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&...
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1answer
37 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
56 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
51 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. ...
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2answers
142 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 ...
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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}[\...
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1answer
47 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 ...
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1answer
48 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 ...
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1answer
63 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 ...
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63 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{...
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1answer
39 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 ...
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1answer
98 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 ...
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$\sum_i X_i^2$ 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 $...
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1answer
55 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
94 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 ...
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2answers
186 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+...
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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 ...
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1answer
61 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 ...
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0answers
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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
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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
50 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 ...
2
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0answers
81 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
33 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 ...