# Questions tagged [conjectures]

for question related to conjectures.

173
questions

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320 views

### Is there an even number $a$ such that $a^{2^{n}}+1$ is prime for infinitely many $n$?

Is there an even number $a$ such that $\{n: a^{2^{n}}+1 \text{ is prime} \}$ is an infinite set?
Let $a$ be even. Is there infinitely many $n$ such that $a^{2^{n}}+1$ is composite?

**0**

votes

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71 views

### A special case of Frankl's conjecture. A question about known results

Let's recall a Frankl's conjecture.
Consider a finite family of finite sets $\mathcal{F}$, such for every pair of sets $A\in \mathcal{F}$ and $B\in\mathcal{F}$, we have $A\cup B\in\mathcal{F}$(the ...

**3**

votes

**0**answers

159 views

### Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures

I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...

**-1**

votes

**1**answer

415 views

### Questions about the abc conjecture [closed]

Question.
Is there an integer $n_0 \geq 2$ such that $$\left\{\frac{c}{rad(abc)^{n_0}}: a, b >0,\; c=a+b,\; \gcd(a, b)=1\right \}$$ is bounded?
The abc conjecture can directly deduce this ...

**4**

votes

**1**answer

184 views

### Lemoine-Lozada circles

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...

**1**

vote

**1**answer

289 views

### Conjecture about primes [closed]

Let $n_0$ be an integer (positive or negative). Are
there infinitely many primes $p$ such that $p + n_0 = {2^r} · q,$ $ r ≥ 0,$ $ q ≥ 3 $ is prime?
When $n_0 = 2$, this conjecture is the twin prime ...

**1**

vote

**0**answers

191 views

### Counting twin primes with a sieve-like algorithm

The sequence A002822, denoted as $S$, represents all the twin primes except $\{3, 5\}$. Other than that exception, $k$ and $k+2$ are twin primes iff $(k+1)/6\in S$. Let $S(N)$ be the subset of $S$ ...

**-4**

votes

**1**answer

331 views

### Are there infinitely many $n$ such that $n!-1$ and $n!+1$ are prime numbers? [closed]

Are there infinitely many $n$ such that $n!−1$ and $n!+1$ are prime numbers?

**12**

votes

**1**answer

3k views

### The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...

**4**

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145 views

### Heuristics for the very little torsion in the cohomology of Shimura variety

Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and A. ...

**5**

votes

**2**answers

264 views

### The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...

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vote

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414 views

### Langlands program and complexity theory

Back when I was an undergraduate, I spent some time reading the about the modularity conjecture, but the details are fuzzy now.
One of the motivations I imagined for the Langlands program was for ...

**17**

votes

**1**answer

2k views

### Conjectures or Results?

There is a paper (not accepted for publication yet) that contains several conjectures. Some of these conjectures were proven recently.
The referee of the original paper requires to substitute the ...

**7**

votes

**3**answers

2k views

### Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple.
I have often heard of the statement that the SZC is stronger than the Riemann ...

**6**

votes

**1**answer

227 views

### Statement of classical Ramanujan-Petersson conjecture

I'm preparing for an expository talk on some topics in the representation theory of reductive p-adic groups, including tempered representations and Whittaker models, and as motivation I wanted to ...

**9**

votes

**1**answer

401 views

### Status of a conjecture of Hirzebruch

I was reading a paper from 1994 which claimed that the following statement was a conjecture of Hirzerbruch:
If a complex surface X is homeomorphic to either $S^2 \times S^2$ or $\mathbb{C}P^2 \# \...

**18**

votes

**3**answers

2k views

### Is it a reasonable way to write a research article assuming truth of a conjecture?

I have found a conjecture in a research article (published in a good journal) on number theory, which is not well known but very reasonable. Let me be clear that, there is no counter-example that vote ...

**18**

votes

**1**answer

2k views

### More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...

**2**

votes

**2**answers

525 views

### On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

This question has been cross-posted from this MSE question and is an offshoot of this other MSE question.
(Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...

**2**

votes

**0**answers

92 views

### A conjectured upper bound for the mean value of prime divisors inside prime gaps

In 1969 C.A. Grimm stated this interesting conjecture: the prime gap $\,G_n=\{x\in N:p_n\lt x\lt p_{n+1}\}\,$ contains at least $\,\#G_n=(p_{n+1}-p_n)-1=g_n-1\,$ distinct prime divisors, that is if $\,...

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vote

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315 views

### Easy to explain conjectures that are still unsolved [duplicate]

Mathematics has many open conjectures which are ridiculously hard to even understand. But this is not always the case. An example is:
Collatz conjecture.
I would like to see some more examples. So ...

**9**

votes

**1**answer

676 views

### Prove that there are no composite integers $n=am+1$ such that $m \ | \ \phi(n)$

Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a<p$ and $ m \ | \ \phi(n)$ then $n$ is prime.
This question is a ...

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vote

**1**answer

299 views

### What groups should I test my conjecture on? [closed]

I have a conjecture that a certain criterion is enough for two groups to be isomorphic. I tested it on all pairs of groups up to size 12, and it worked like a charm. I know, however, that groups are ...

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393 views

### Collatz conjecture in all its variants

There are all kinds of execution variants to the collatz conjecture for when hitting an odd number:
$3n+1$ or $3n+3^a$ or $1.5n + 0.5$ or $1.5n + 1.5$... . The assumption is: proving any of them will ...

**5**

votes

**1**answer

224 views

### Inductively computing Mersenne primes / perfect numbers?

For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$.
Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$.
Set $\hat{\phi}(1) = \{x_1\}$ and ...

**9**

votes

**1**answer

353 views

### $π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.)
Thinking about the prime number theorem, I wondered whether it is ...

**4**

votes

**1**answer

329 views

### The action of the unitary divisors group on the set of divisors and odd perfect numbers

Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}\mid d^2 \...

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392 views

### Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...

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votes

**2**answers

289 views

### Conjecture about minimal number of edge crossings in complete bipartite graphs

I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$.
The Wikipedia article https://en....

**0**

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**1**answer

172 views

### "Halfway" approach to Landau's 4th problem (special case of Bateman-Horn)

Landau's 4th problem asks if $n^2 + 1$ is prime for infinitely many $n \in \Bbb{Z}$. It is known that $n^2 + 1$ can only be divisible by Pythagorean primes, and that for any $p$ congruent to $1 \pmod ...

**17**

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747 views

### Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,z\in\mathbb Z$

Roger Heath-Brown conjectured that any integer $k\not\equiv\pm4\pmod9$ can be expressed as $x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$ in infinitely many different ways. Also it is well-known that some ...

**6**

votes

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938 views

### Is there any relationship between Szemerédi's theorem and Sunflower conjecture?

I have observed some similar things between a reformulation of the Sunflower conjecture (see also conjecture 1.3 in Improved bounds for the sunflower lemma) and Szemerédi's theorem such that for ...

**8**

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**1**answer

993 views

### Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...

**9**

votes

**1**answer

393 views

### A constant bizarrely related to the Fibonacci Numbers

For roughly the past month, I have been studying denesting radicals. For example: the expression $\sqrt[3]{\sqrt[3]2-1}$ is a radical expression that contains another radical expression, so this ...

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**0**answers

275 views

### A generalization of Feit–Thompson conjecture, for square-free integers

I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...

**4**

votes

**1**answer

308 views

### About semiprimal representations of $1$

Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...

**0**

votes

**1**answer

439 views

### Has Pillai conjecture been proven?

I found the paper https://hal.archives-ouvertes.fr/hal-00698687v9/document which claims the proof of Pillai conjecture.
However, it is not mentioned anywhere that it has been proved. It's stated ...

**3**

votes

**0**answers

370 views

### Equivalence of the union-closed sets conjecture that is locally weaker of any use?

Let $F$ be a union-closed family. We call $F$ minimal if for every $x\in \cup(F)$ we find $S\in F$ such that $S\backslash \{x\} \in F$. It is sufficient to proof the union-closed sets conjecture for ...

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55 views

### Schur positive expression involving border-strip tableaux

Recall the power-sum expansion of Schur functions,
$$
s_\lambda = \sum_\mu \chi^{\lambda}(\mu) \frac{p_\mu}{z_\mu},
$$
in terms of Sn-character. These can be calculated
by the Murnaghan-Nakayama rule, ...

**9**

votes

**1**answer

382 views

### Conjectured primality test for specific class of $N=4kp^n+1$

Can you provide a proof or counterexample for the following claim?
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...

**6**

votes

**2**answers

673 views

### Conjectures and open problems in representation theory [closed]

Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?

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96 views

### Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function

The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...

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votes

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582 views

### is there a link with the probabilistic model for prime numbers?

Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...

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votes

**1**answer

749 views

### Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...

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votes

**1**answer

341 views

### Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture

I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia Firoozbakht's conjecture.
Question. What articles have been published in ...

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votes

**1**answer

155 views

### On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture

I wondered, inspired in a result from [1] (Proposition 17) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,
$$\sum_{\substack{\text{...

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votes

**1**answer

236 views

### Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.
If the ...

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vote

**0**answers

41 views

### On quasiperfect numbers, $\operatorname{rad}(\sigma(n))=\operatorname{rad}(2n+1)$ and $\operatorname{rad}(\sigma(n)-1)=\operatorname{rad}(2n)$

An integer $n\geq 1$ is said quasiperfect number if the sum of its positive divisors $\sigma(n)$ is equal to $2n+1$. See the Wikipedia Quasiperfect number.
The idea of this post is ask about the ...

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votes

**1**answer

394 views

### A generalization of Lander, Parkin, and Selfridge conjecture

My question: Are the conjectures as follows correct?
Given a positive integer $P>1$, let its prime factorization be written
$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}...p_k^{a_k}$$.
Define the ...

**8**

votes

**3**answers

1k views

### Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...