Questions tagged [conjectures]
for question related to conjectures.
198
questions
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Is there any progress toward solving Gilbreath's conjecture?
Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the ...
10
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2
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983
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Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that we have
$$\sup ...
1
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0
answers
138
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Optimal covering trails for every $k$-dimensional cubic lattice $\mathbb{N}^k := \{(x_1, x_2, \dots, x_n) : x_i \in \mathbb{N} \wedge n \geq 3\}$
After a dozen years spent investigating this particular class of problems, I finally give up and I wish to ask you if any improvement is achievable from here on.
The general problem is as follows:
Let ...
3
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4
answers
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A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...
7
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0
answers
580
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What will be the consequences if second Hardy-Littlewood conjecture turns out to be true?
It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is,
What would be the consequences if Second Hardy-Littlewood ...
3
votes
0
answers
211
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Uses of excluded middle on a conjecture that can be rewritten constructively with this trick
An interesting proof technique is to use the law of excluded middle on a conjecture. There are proofs using LEM on the Riemann hypothesis for example.
Constructively this is disallowed (if you can ...
8
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0
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431
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Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
7
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1
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493
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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 \...
20
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1
answer
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Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...
1
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0
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Initial conditions to falsify Rowland's conjecture
Based on the Rowland's paper (A natural prime-generating recurrence), is there any theorem to show that for which initial condition $a(1) = k$ the conjecture can be falsified?
For example, for $k$ ...
3
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1
answer
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What do we know about Lucky numbers?
I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics.
Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
6
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1
answer
319
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Different flavours of Vassiliev Conjecture
There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...
11
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1
answer
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Seeking proof of the Cuckoo Cycle Conjecture
Allow me to introduce the Cuckoo Cycle Proof of Work, and conclude with a closely related Conjecture about random graphs, which I hope someone can find a proof for.
Cuckoo Cycle is named after the ...
2
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2
answers
474
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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 ...
12
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2
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516
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Sylvester–Gallai theorem with circle version, plane version and curve version?
The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
All the points are collinear; or
There is a line which contains exactly two of the ...
3
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0
answers
355
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A conjecture on consistent monotone sequences of polynomials in Bernstein form
A Conjecture
In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...
2
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0
answers
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On a subset of the $abc$ triples
The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and
$$c > \...
1
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0
answers
312
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How would one go about solving this conjecture concerning exponential Diophantine equations?
I’ve been working on the Collatz Conjecture, and I believe I’ve reduced it to a more tractable problem. Unless there are some errors I’ve overlooked, I have managed to reduce the Collatz Conjecture to ...
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3
answers
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Maximum matching in a graph with no "shortcuts"
For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^...
2
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0
answers
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Conjecture about primes and Fibonacci numbers
I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's ever possible to find a ...
13
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1
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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. ...
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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,...
1
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1
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Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers
Let $k \in \mathbb{Z}^+$.
Is it possible to prove that, for some given
$m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$,
there are only finitely many $k$ such that the closed ...
15
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0
answers
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Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
0
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0
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Conjecture on the unsolvability of the $\{3 \times 3 \times \cdots \times 3\} \subseteq \mathbb{R}^k$ dots problem starting from the central point
In 2020 (see Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm, JFMA, 3(2), p. 96), I conjectured that, in the Euclidean space $\mathbb{R}^k$, we can cover any given set of ...
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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 $...
17
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5
answers
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Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$
in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?
9
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2
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571
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Strengthened version of Isoperimetric inequality with n-polygon
Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
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0
answers
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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 ...
0
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0
answers
82
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Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$
Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia)
$$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...
2
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0
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285
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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 =...
6
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1
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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 ...
12
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2
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902
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An analogue of the Bass-Quillen conjecture with power or Laurent series
The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
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0
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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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1
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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 ...
2
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1
answer
296
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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
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1
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586
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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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3
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Prime constellation conjectures
This is a simple question about terminology and provenance.
I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.
I've encountered Polignac's ...
2
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0
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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-...
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3
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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 ...
23
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1
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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 ...
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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 ...
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9
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Mathematical conjectures on which applications depend
What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
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Total coloring conjecture for Cayley graphs
The total coloring conjecture (TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree ...
10
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1
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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 ...
5
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1
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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
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0
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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 ...
5
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0
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313
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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 ...
6
votes
1
answer
279
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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 ...
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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 ...