Questions tagged [conjectures]

for question related to conjectures.

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Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$

I try here because I expect I cannot have any answer on MSE : Problem : Let : $$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$ Then it seems $\exists y\in(0,1)$ and $...
3 votes
0 answers
291 views

A conjecture on consistent monotone sequences of polynomials in Bernstein form

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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Stronger conjectured inequality for area of a polygon

Four years ago, I proposed an inequality related to area and sides of a polygon. After computer checking, I conjecture that the previous inequality can be strengthened as follows: Let $A_1A_2\cdots ...
2 votes
0 answers
246 views

How soon can we represent a number as the sum of two primes?

Posting in MO since it was unanswered in MSE. Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
0 votes
0 answers
149 views

Does Goldbach's conjecture imply this conjecture?

Does Goldbach's conjecture imply this conjecture: For any $x$, there exists an $n$ such that the number of goldbach partitions of any even integer $\geq n$ is greater than $x$?
2 votes
1 answer
273 views

Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
20 votes
1 answer
551 views

Distinct exponents in the factorization of the factorial, a problem of Erdős

In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...
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2 votes
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A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?

The definitions are from these two questions: https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers https://math.stackexchange.com/questions/4349186/trying-to-understand-...
12 votes
3 answers
557 views

An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
2 votes
3 answers
406 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
9 votes
1 answer
330 views

A counterexample to a conjecture of Lawson

Yau quotes Lawson as having formulated the following conjecture [1]: Let $M$ be an embedded minimal surface in $\mathbf{S}^3$. Prove that the two domains in $\mathbf{S}^3$ divided by $M$ have equal ...
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5 votes
1 answer
350 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?
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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 ...
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5 votes
0 answers
252 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
465 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 ...
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5 votes
1 answer
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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
310 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 ...
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3 votes
0 answers
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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
415 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?
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12 votes
1 answer
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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 votes
0 answers
170 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. ...
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6 votes
2 answers
370 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 ...
1 vote
0 answers
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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
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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 ...
8 votes
2 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
367 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
438 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 \# \...
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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 ...
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23 votes
1 answer
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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
550 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
98 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 $\,...
1 vote
2 answers
415 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
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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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1 vote
1 answer
309 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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1 vote
0 answers
935 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 ...
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5 votes
1 answer
230 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 ...
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9 votes
1 answer
368 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 ...
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4 votes
1 answer
343 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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5 votes
0 answers
470 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 $\...
5 votes
2 answers
316 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....
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1 answer
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"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 votes
0 answers
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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 ...
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6 votes
1 answer
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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 ...
9 votes
1 answer
1k 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
405 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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8 votes
0 answers
297 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 ...
5 votes
1 answer
379 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 \...
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0 votes
1 answer
553 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
380 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 ...
3 votes
0 answers
65 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, ...