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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0answers
87 views

Are 2, 3 the only prime numbers that don't have the digit 1 and are palindromes whose squares are also palindromes?

While thinking about prime numbers, I noticed that: $(1)$ Very few prime numbers have squares that are palindromes. Ex: $2$, $3$, $11$, $101$, $307$ $(2)$ Even rarer are prime numbers that are ...
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72 views

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

While discussing prime powers and divisors, I came up with the following problem. Examples $\to$ prime $p=3$ digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,. $\to$ prime $p=7$ ...
5
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1answer
104 views

Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
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1answer
104 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+3+6 = ...
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1answer
202 views

Is $\zeta(s)\sim\sqrt{\frac{\zeta(4s)}{\zeta(2s)}}\prod\limits_{n=1}^\infty\big(1-\frac{2}{p_n^s+p_n^{-s}}\big)^{-1/2}$?

The Riemann Zeta function, denoted by $\zeta(\cdot)$, is defined by the following equation for $s > 1$ and $p_n$ the $n^\text{th}$ prime number. $$\zeta(s)=\prod_{n=1}^\infty\bigg(1-\frac{1}{p_n^s}\...
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1answer
88 views

How does one subtract from concatenation in order to prove that $4\times 5 + 67 = 45 + 6\times 7$?

I noticed that if we get the numbers $4$, $5$, $6$ and $7$, they have an interesting property! $$4 \times 5 + 67 = 45 + 6 \times 7\tag*{= 87.}$$ I then conjectured that these were the only four ...
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31 views

Conjecture on the growth of $q_1 = 1, q_{n+1}=q_n + f(q_n) $

This is a generalization of my answer to Calculate the limit of the following recurrent series in the form suggested by Will Jagy. $q_1 = 1, q_{n+1}=q_n + f(q_n) $ where $f(x) > 0$ and $f'(x) <...
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1answer
35 views

On a method to solve certain recursive sequences - looking for counterexamples?

When I started with this question, I wanted to know why my reasoning was wrong. Nevertheless, after checking some examples, I've noticed that my conjecture was actually - or at least seems to be - ...
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2answers
52 views

If $q$ is a Fermat prime, is $\sigma(q^k)/2$ a square if $k \equiv 1 \pmod 4$?

In what follows, let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number of the form $M = 2^{2^m} + 1$ is called a Fermat number. If in addition $M$ is prime, then $M$ is ...
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177 views

Constraints for the number of basis sets vs number of sets for a union-closed sets conjecture counterexample

I searched some literature about constraints on the number of basis sets (or $\cup$-irreducible sets, also called generators) $\vert\mathit{J}(\mathcal{F})\vert$ with respect to the number of sets $\...
2
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1answer
26 views

On a conjecture that two intersections are at $\big(\pm\frac{S}m, S\big)$ (about linear equations)

Suppose we have two linear equations, $y=mx$ and $y=-mx$ , inserted on the same plane for $m>0$. These equations are going to have an intersection at $(0,0)$. Now, let $$\begin{align}S&=1+\sum_{...
1
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1answer
75 views

Symmetry conditions for symmetric random vectors

While formulating the properties for a certain statistical model I'm dealing with, I came up with the following question (with credit going to MikeEarnest in comments for the proper formulation). A ...
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1answer
54 views

Sum on GCD and prime numbers

I was studying gcd then I encountered this sum $(1).$ A conjecture: If $(1)=1$ for any values of $N\ge3$, then N is a prime number. Let: $$f(N)=\frac{1}{N^{1-s}(N-1)}\sum_{j=1}^{N}(-1)^jj^s\frac{{...
5
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0answers
110 views

Conjecture about primes and the greatest common divisor

Conjecture: Given $m,n\in\mathbb N^+$, one odd and one even, there are two primes $p,q$ such that $|mp-nq|=\gcd(m,n)$. I hope MSE can determine its validity. From time to time, when testing my ...
3
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2answers
91 views

On a conjecture that $\sum_{i=1}\limits^na_ip_i = 1\Rightarrow \exists\big(\sum_{i=1}\limits^na_i=0\big)$ (about primes)

Conjecture: Denote by $p_i$ the $i^{\text{th}}$ prime number; by $a_i$ the $i^\text{th}$ arbitrary integer; and by $\exists(x)$ the existence of a chosen $x$.$$\sum_{i=1}^na_ip_i = 1\Rightarrow \...
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2answers
126 views

An explicit “formula” for the prime counting function?

It is known that $\log(p_1),\cdots,\log(p_n)$ are linearly independet over $\mathbb{Q}$, where $p_i$ denotes the $i$-th prime. For a number $1 \le k \le n$ let $Log(k)$ denote the vector with respect ...
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1answer
88 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
93 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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152 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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55 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
228 views

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

(Following question 2269073. See also respective lower bounds.) 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{...
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1answer
185 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
248 views

Is This a New Property I Have Found Pertaining to 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
3k 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} ...
28
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3answers
921 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 values $a,...
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36 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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0answers
89 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 ...
2
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3answers
119 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
160 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
211 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
94 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 ...
47
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0answers
649 views

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
81 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
84 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 ...
2
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1answer
66 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\...
4
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2answers
64 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
238 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
64 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{...
1
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1answer
82 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
43 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
102 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
68 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
160 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
88 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
58 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
67 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
80 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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0answers
102 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
57 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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