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Questions tagged [conjectures]

Use this tag if your question is about a well-known conjecture or a conjecture of your own.

3
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1answer
35 views

Free action of finite cyclic group on $S^3$

I'm starting to read Allen Hatcher's Space of Knots and at the end of the first paragraph he says that his work would apply to all knots if the following conjecture is true: Every free action of a ...
-2
votes
0answers
59 views

$ \pi( x^v + v \space x) - \pi( x^v + 4) < \pi(x+4) + v $ ( Tommy’s note )

In tommy’s very old notebook I found [1] $$ \pi( x^v + v \space x) - \pi( x^v + 4) < \pi(x+4) + v $$ Where $\pi$ is the prime counting function , $x$ is an integer larger than $ v + 4 $ and $ v $...
3
votes
1answer
81 views

A conjecture on the closeness of twin primes

Let $p_1$ and $p_2$ be twin primes, and let $p_1-1=a_1\times b_1$ and $p_2+1=a_2\times b_2$ be such that $|b_1-a_1|$ and $|b_2-a_2|$ are minimised. Similarly, let $p_1+1=p_2-1=a\times b$ be such that ...
0
votes
1answer
43 views

Subsubgroups are subgroups of subgroups / Multiplicative Property of the Index

Algebra by Michael Artin Prop 2.8.14 Multiplicative Property of the Index Statement of Prop 2.8.14 Let $G \supseteq H \supseteq K$ be subgroups of a group G. Then $[G:K] = [G:H][H:K]$. Proof of ...
2
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0answers
46 views

Inquiry about the Wolfram MathWorld entry on Odd Perfect Number

I just have a quick inquiry about the Wolfram MathWorld entry on Odd Perfect Number. Last sentence of the fifth paragraph states that: Hagis (1980) showed that odd perfect numbers must have at ...
1
vote
1answer
65 views

There are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$

Conjecture: Given $a,b\in\mathbb Z^+$ there are primes $p,q$ such that $\gcd(a,b)=|ap-bq|$. I would like help with a proof or a counter-example. Tested for millions of pseudo random numbers.
1
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0answers
77 views

A problem inspired by the Twin Prime conjecture

I came up with a question several hours ago...but I couldn't find any information about it. The problem goes like below $$P^n_k =\{(p_1,...,p_n)|p_1<...<p_n:primes,p_n-p_1\le k\}$$ $$k_n=min\{...
2
votes
0answers
65 views

Prime twins and $ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $

Let $p$ be a prime such that $p+2$ is Also a prime. Define $$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$ For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always ...
0
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4answers
82 views

A conjecture of exercise type: $\gcd(a,b)^2=\gcd(a^2+b^2,ab)$

This must be known, but I haven't found it and want help to prove it: For all $a,b\in \mathbb Z$, $\gcd(a,b)^2=\gcd(a^2+b^2,ab)$ Tested for $10,000,000$ pseudo random number pairs.
3
votes
2answers
64 views

A confession and a conjecture $\gcd(a-b,a+b)|2\gcd(a,b)$

A long time ago I studied mathematics at the University of Stockholm. I was mostly interested in algebra and topology but as everybody interested in math I was fascinated by the most famous problems ...
3
votes
3answers
65 views

Conjecture: $ \gcd(\operatorname{rad}(a+b) ,ab)= \operatorname{rad}(\gcd(a,b))$

I have discovered some exercise type conjectures which I can't prove and this is one of them: Given positive integers $a,b$, then $$ \gcd(\operatorname{rad}(a+b) ,ab)= \operatorname{rad}(\gcd(a,b)...
8
votes
4answers
113 views

Conjecture that $ \frac{\gcd(a+b,ab)}{\gcd(a,b)} \mid \gcd(a,b)$

I have discovered some exercise type conjectures which I can't prove and this is one of them: Given positive integers $a,b$, then $$ \frac{\gcd(a+b,ab)}{\gcd(a,b)}\ \bigg|\ \gcd(a,b)$$ Can this ...
3
votes
0answers
85 views

Question about a result on odd perfect numbers

In the paper titled Improving the Chen and Chen result for odd perfect numbers (Lemma 8, page 7), Broughan et al. show that if $$\frac{\sigma(n^2)}{q^k}$$ is a square, where $\sigma(x)$ is the sum of ...
0
votes
0answers
28 views

Generalization of Legendre's conjecture

I suggest the following is true. I've verified it for small values with Mathematica and heuristically it seems solid. For $n \ge p^{k}$ with prime $p \ge 5$, there are primes $r_i > p$ such that $...
88
votes
11answers
13k views

Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?

In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object,...
1
vote
1answer
115 views

Is it possible to narrow down a domain of possible counter-examples to the Collatz Conjecture?

First of all, I am not trying to prove the Collatz Conjecture. I want to know if it is possible to rule out certain values of a counter-example. Suppose $k \in \Bbb Z^+$ is the lowest counter-...
1
vote
1answer
36 views

Understanding “A note on the union-closed sets conjecture”

I am trying to understand the article "A note on the union-closed sets conjecture" by Roberts and Simpson. At the first paragraph of the second page they write: "We say that $A$ is a basis set in $\...
5
votes
1answer
76 views

Positive integers have the form $a^2+b^2+c^2+2^d$

For any positive integer $n$ there seems to be non-negative integers $a,b,c,d$ such that $$n=a^2+b^2+c^2+2^d.$$ Due to Legendre's three-square theorem a natural number can be represented as the ...
0
votes
1answer
36 views

On the quantity ${n^2}/D(n^2)$ where $n^2$ is the non-Euler part of members of the OEIS sequence A228059

Let $\sigma(x)$ denote the sum of the divisors of the number $x \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $x$ as $D(x):=2x-\sigma(x)$. This afternoon I noticed some ...
1
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0answers
32 views

Specific evidence for or against the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd ...
3
votes
2answers
186 views

Show that $\int_0^1 4 \space\operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $

My mentor tommy1729 wrote $\int_0^1 4 \space \operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $ I wanted to prove it thus I looked at some methods for computing integrals and also ...
1
vote
1answer
31 views

Conjecture about special grid of numbers

Consider you have created grid of numbers like following image starting from any positive integer (in this case 8) To create such grid, follow this steps; Pick a number greater than one and write ...
0
votes
0answers
21 views

Is ${n^2}/D(n^2) \in \mathbb{N}$, if $q^k n^2$ is an odd perfect number?

Let $x \in \mathbb{N}$, the set of positive integers. The sum of the divisors of $x$ is denoted by $\sigma(x)$. Denote the deficiency of $x$ by $D(x):=2x-\sigma(x)$, and the sum of the aliquot parts ...
7
votes
0answers
59 views

Conjectures for which there are strong heuristic arguments both for and against

I find the conjecture that there are infinitely many Fermat primes to be very interesting, because there is a very reasonable-seeming and semi-quantitative heuristic argument for the conjecture, but ...
21
votes
2answers
444 views

Prove that $\sum\limits_{i=1}^{n} a_i\geq n^2$.

A hint can be helpful, but not a whole solution. The Problem (conjecture): Given a natural number $n \geq 1$ and a sequence of natural numbers $(a_i)_{1 \leq i \leq n}$ in which for every pair $(i,...
2
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0answers
59 views

On Sierpinski's Conjecture

The statement of Sierpinski's Conjecture is 'for every integer n>1, there exist three integers a,b,c such that 5/n=(1/a)+(1/b)+(1/c)' Would this conjecture be proved if we could show this for every ...
2
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0answers
227 views

What is known about the $7x+1$ problem?

One of the most famous problems in mathematics that remains unsolved is the Collatz conjecture. I am concerned with similar 7x+1 problem. I have already seen this problem mentioned in the literature....
2
votes
1answer
112 views

The equation $\varphi(n)=n-\log_2(1+\operatorname{gpf}(n))-\operatorname{gpf}(n)+1$ and Mersenne primes

Let $n\geq 1$ an integer, we denote the Euler's totient function as $\varphi(n)$ and the greatest prime dividing $n$ as $\operatorname{gpf}(n)$ (that it the arithmetic function defined in the ...
1
vote
1answer
95 views

A modular conjecture about odd primes [closed]

Conjecture: $n^{2p-1}\equiv n\pmod {2p}$ for all $n\in \mathbb N$ and all odd primes $p$. I started investigate the least $x_n$ such that $p_n^{p_{n+1}}\equiv x_n\pmod{p_{n+2}}$ and ended up with ...
10
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2answers
114 views

If $a+b+c$ divides the product $abc$, then is $(a,b,c)$ a Pythagorean Triple?

Firstly, I will define what Pythagorean Triples are for those who do not know. Definition: A Pythagorean Triple is a group of three integers $a$, $b$ and $c$ such that $a^2+b^2=c^2$, ...
0
votes
1answer
157 views

Conjecture about the prime number function

Define $p_{n,m}$ by $p_{n,1}=p_n$ and $p_{n,m}=p_{p_{n,m-1}}$, where $p_n$ is the prime number function. Conjecture: $n^n<p_{n,n}< (n+1)^{(n+1)}$ I've only tested it for $n=1,\dots,11.$ I ...
10
votes
0answers
247 views

A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\...
2
votes
1answer
118 views

Generalization of Opperman's Conjecture

Does this conjecture have a name? What about a counterexample?: $$ \forall n,k \in \mathbb{N}, k \gt 1, \exists d \in (kn-n,kn] \text{ s.t. } d \perp n! $$ An equivalent statement is this: Take a ...
4
votes
1answer
54 views

A conjectured formula for the polylogarithm of a negative integer order

I discovered the following formula while working on the sequence A141697 from the OEIS. I have no idea whether it is something trivial or not. I would be very happy to know more about it. $$ \textrm{...
3
votes
0answers
81 views

A conjecture regarding a strange infinite sum

Numerical calculations indicate that the following conjecture is true $$ \sum _{n=0}^{\infty} \frac{\prod _{k=0}^{n-1} \left(x+e^{-k}\right)}{\prod _{k=0}^n \left(1+te^{-k}\right)}t^n =\frac{1}{1-xt}.\...
2
votes
2answers
158 views

Trigamma identity $\psi_1\left(\frac{11}{12}\right)-\psi_1\left(\frac{5}{12}\right)=4\sqrt 3 \pi^2-80G$

Hello regarding this integral: Integral $\int_0^1 \frac{\sqrt x \ln x} {x^2 - x+1}dx$ The following conjecture comes: $$\psi_1\left(\frac{11}{12}\right)-\psi_1\left(\frac{5}{12}\right)=4\sqrt 3 \pi^2-...
1
vote
4answers
77 views

On a conjecture that $\sum\limits_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow} 2$.

I have made the following conjecture, and I do not know if this is true. Conjecture: \begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| ...
1
vote
1answer
53 views

Approaches to Brocard's problem

The brocard problem is an unsolved problem that asks how many integers m and n exist such that $n! + 1 = m²$ More specifically it conjectures there are only three such numbers. What are some ...
4
votes
1answer
48 views

On a conjecture that $P_n^{\,2}+5^2+2^k=(P_n-1)^2+l^2$.

I was looking at perfect numbers and came across something that might serve a little interesting. Denote by $P_n$ the $n^\text{th}$ perfect number, then there appears to always exist $k\in\mathbb{W}...
6
votes
1answer
91 views

Conways 99 graph problem

The Conway 99 graph problem is stated here as one of 5 problems. My question is this: Is there somewhere a list of examples with "Conway graphs" with fewer then 99 vertices? Or can you provide ...
2
votes
0answers
57 views

Is $\sum_{i=1}^{n-1}i^{n-1}\equiv -1\pmod {2n}\Leftrightarrow \text{$n$ is prime }\equiv 3\pmod 4$?

I was looking at the Agoh-Giuga Conjecture, namely, $$\sum_{i=1}^{n-1}i^{n-1}\equiv-1\pmod n\Leftrightarrow n\text{ is prime.}\tag1$$ I decided to see if I could prove it, expecting lots of ...
2
votes
0answers
114 views

Conjectured inequality: $\frac{p_{2n+1}-1}{2}\geqslant p_{n+1}+2(n+1)^{2/(n-1)}+\frac12\log_{4}^2(n-6)\qquad(n>6)$

I was inspired by this post to look at some prime numbers and see if I can find a bound. In the post, a conjecture was made with variables $m$ and $n$ such that $(m,n)\neq (1,1)$ respectively. I was ...
11
votes
1answer
107 views

Does the prime sequence satisfy $p_n+p_m \le p_{n+m} < p_n p_m$?

I did some experiments in SAGE and it seems like the prime sequence $p_n$ satisfies: $$p_n+p_m \le p_{n+m} < p_n p_m$$ for $(n,m) \neq (1,1)$. For $n=1$ the last inequality is Bertrands ...
6
votes
1answer
58 views

Is my proof correct on how $k$ must be a power of $2$? Are there other proofs?

So I was looking at the Fermat Primes. These are primes of the form $2^k+1$ for a natural number $k$, such that I define by $\mathbb{N}:=\big\{1,2,3,\ldots\big\}$ and $0\notin \mathbb{N}$. We denote ...
1
vote
4answers
90 views

Is it true that $2^{2^{n}}$ always ends in the digit $6$?

As the question says, is it true, that for $n >1$, $2^{2^{n}}$ ends in the digit 6? How would one prove this? It seemed true. I considered writing it as $$2^{2^{n}} = \prod_{k=0}^{n} 2^{ n\choose ...
0
votes
0answers
15 views

What is the analogy between the plane and function fields?

This question arised when I was thinking about the Lonely Runner Conjecture. The LRC is an example of a Conjecture which we can 'rewrite' in terms of function fields, where we use that $\mathbb{Z}$ is ...
6
votes
2answers
134 views

Is this integral $\int_0^\infty\frac{\cos(a x+ 2b \arctan x)}{x^2+1}dx$ exactly zero when $b\in\mathbb{N}$?

I recently encountered this integral $$\int_0^\infty\frac{\cos(a x+ 2b \arctan x)}{x^2+1}dx$$ which suspiciously close to 0 for nonzero integer values of $b$, as indicated by numerical calculations. ...
1
vote
0answers
78 views

Is $N^n(p_n\#)-p_n\#$ always prime for all $n\in\mathbb{N}_{>1}$?

I have made a conjecture, and it seems like this can generate prime numbers very very well and in their oder too. Conjecture: Consider the following primorial for $n\in\mathbb{N}$: $$...
0
votes
1answer
47 views

Do odd numbers $n$ satisfying $\gcd(n, \sigma(n)) > \sqrt{n}$ have a special form?

Let $\sigma(n)$ denote the sum of divisors of the positive integer $n$. Using Sage Cell Server, I was able to get the following odd numbers $n < 5000$ satisfying $\gcd(n, \sigma(n)) > \sqrt{n}$:...
2
votes
1answer
58 views

Confirmation of Proof: $\pi_n + p_a + p_b \geqslant \sum_{i=1}^4 x_i$.

I was messing around with numbers and I made the following conjecture: Conjecture: Let $\pi_n$ be the $n^{\text{th}}$ perfect number; $p_a$ be the prime after $\pi_n$ and $p_b$ be the ...