# Questions tagged [conjectures]

for question related to conjectures.

196
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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 $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...

20
votes

1
answer

812
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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
vote

0
answers

152
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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 $k$ ...

6
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1
answer

284
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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 ...

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0
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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

290
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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 ...

2
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0
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236
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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 ...

3
votes

1
answer

339
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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 ...

1
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1
answer

171
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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
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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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1
answer

172
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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
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0
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354
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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 ...

1
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0
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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
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0
answers

268
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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 =...

2
votes

1
answer

293
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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
answer

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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{...

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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636
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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 ...

2
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2
answers

468
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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 ...

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1
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445
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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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376
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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?

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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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303
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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 ...

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1
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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 ...

6
votes

1
answer

271
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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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1
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315
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### 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 ...

3
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0
answers

233
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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 $S$ ...

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1
answer

613
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### 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?

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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### 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. ...

6
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2
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449
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### 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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0
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515
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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 ...

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1
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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 ...

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2
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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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1
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607
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### 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
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1
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### 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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3
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### 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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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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2
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622
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### 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
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0
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### 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 $\,...

2
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2
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### 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
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1
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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
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340
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### 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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### 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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### 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
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1
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383
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### $π(x+y) - π(x) ≤ c·y/\ln(y)$ 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
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1
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382
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### 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 \...

6
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0
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496
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### 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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2
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357
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### 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....