# Questions tagged [conjectures]

for question related to conjectures.

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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 $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. ...
812 views

### 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
152 views

### 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 $k$ ...
284 views

### 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 ...
1 vote
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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-...
636 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 ...
468 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 ...
445 views

### A counterexample to a conjecture of Lawson

Yau quotes Lawson as having formulated the following conjecture : 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 ...
376 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 ...
303 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 ...
502 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 ...
271 views

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
315 views

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 ...
233 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 $S$ ...
613 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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### 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. ...
181 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. ...
449 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
515 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 ...
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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 ...
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 ...
607 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 ...
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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 ...
817 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 ...
1 vote
340 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 ...
1 vote
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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 ...
236 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 ... 383 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 ...
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 $S_n=\{d \in \mathbb{N}\mid d^2 \... 6 votes 0 answers 496 views ### Does the equation$\sigma(\sigma(x^2))=2x\sigma(x)$have any odd solutions? This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered. Let$\...
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....