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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.

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Which numbers will iterate to others under the Collatz iteration?

I have a question about the Collatz conjecture and how some numbers merge trajectories. Take the standard map: $$C(n) = \begin{cases} n/2, & \text{if $n \equiv 0$ mod $2$} \\ 3n+1, & \text{...
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1answer
33 views

A weaker conjecture than a known conjecture

I really apologize if my question is not appropriate here, though I hope it is. Let $C$ be any known conjecture in mathematics, which is still open. Let $D$ be another conjecture such that a positive ...
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1answer
66 views

Where is the flaw in this proof of Legendre's Conjecture?

Introduction The following argument has been advanced by one of my friends which attempts to prove the Legendre's Conjecture. I could find no flaw in the argument and so I am posting it here in the ...
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0answers
24 views

$\sum_i X_i^2$ has $\chi^2_{n}$ distribution and $X_i$ i.i.d. imply $X_i$ normal

Let $X_1,\ldots,X_n$ be i.i.d. random variables with distribution $F$. It is known that if $F$ is the standard normal distribution then $$ S:=\sum_{i=1}^n X_i^2 $$ has a chi square distribution with $...
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1answer
51 views

When given a non-multiple of $3$, $k$, is it possible construct $m<k$ with these conditions? [on hold]

This is another Collatz-related problem about trying to represent a number in a certain form. As is usually the case with the Collatz conjecture, this is probably not useful. My question is : Can ...
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0answers
39 views

How to prove this conjecture: Prime Integer Partitions

Conjecture: every prime number that is equal or greater than 271 has a prime integer partition (PIP). For an integer partition of n to be a PIP, it is necessary that: n must be prime All the terms ...
4
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1answer
77 views

If $x$ and $y$ are solitary numbers satisfying $\gcd(x,y)=1$, under what conditions does it follow that $xy$ is also solitary?

Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$. Denote the abundancy index of $z$ by $I(z) = \sigma(z)/z$. If the equation $I(z)=I(a)$ has the lone solution $z=a$, then $a$ is said ...
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2answers
174 views

Can any integer, not a multiple of three, be represented as $n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$?

This question has some relevance to the Collatz conjecture. It was originally based on trying to represent a number like this: Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+...
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3answers
50 views

Divisibility rule

Example: $2^1$=2 --> $2\mid2$ If a number has their last digit divisible by 2, than the number is divisible by 2 $2^2$=4--> $4\mid2$, $4\mid4$ If a number has their last two digit divisible by ...
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1answer
50 views

Can this bound for the abundancy index of $n$ be improved, given that $q^k n^2$ is an odd perfect number with $k=1$?

In what follows, set $I(x)=\sigma(x)/x$ to be the abundancy index of $x \in \mathbb{N}$, where $\sigma(x)$ is the sum of divisors of $x$. If $I(y)=2$ and $y$ is odd, then $y$ is called an odd perfect ...
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0answers
48 views

Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
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0answers
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Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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2answers
43 views

Unique factorization conjecture?

Let $A_p$ be an Integral domain. Conjecture : If every $a$ in $A_p$ that equals $b \space c$ for irreducible elements $b,c$ in $A_p$ , has Unique factorization then the Integral domain $A_p$ is a ...
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0answers
61 views

Fortune's conjecture solved for limited cases?

I am not a mathematician, but while doing other work, I came across the Fortune conjecture. According to Wikipedia and other research, it seems that it has not yet been proven. I thought about the ...
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0answers
29 views

Schwarz inequality for unital positive maps on C*-algebras

I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that Let $\phi : \mathcal{A} \to \mathcal{B}$ be a unital positive linear ...
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1answer
50 views

Validation for a conjecture about Chinese Remainder Theorem for groups

I was wondering if the following statement is true: Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $i\neq j$. Then $G/H_1\cap H_2...\cap H_n\cong G/H_1 \times......
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2answers
52 views

Conjecture for $\ln(x)$ and $\frac{d^ny}{dx^n}$

I'm not really sure where to start, I found the first, second, third and fourth derivatives of $\ln(x)$ to be $\frac{1}{x}$,-$\frac{1}{x^2}$, $\frac{2}{x^3}$, and -$\frac{6}{x^4}$, respectively. ...
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1answer
94 views

Calculating the $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}$

Question: If $s \in \mathbb{N}$ is it true: $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}={\zeta\left[{s+1 \choose 2}\right] \above 1.5pt \prod_{k=1}^s{s \choose k}};$ where $\...
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2answers
65 views

Primes of the form $p_1p_2\dotsm p_k+1$ [closed]

Let $p_1,p_2,...,p_k$ be the first $k$ primes and $k>1$. Does there exist a prime $p_{k+1}$ such that $p_{k+1} = p_1p_2...p_k + 1$?
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1answer
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What is the common (and simplified) value for $D(q^k)D(n^2) = 2s(q^k)s(n^2)$ when $q^k n^2$ is an odd perfect number?

In an answer to an earlier question, it is shown that $$D(2^p - 1)D(2^{p-1}) = 2s(2^p - 1)s(2^{p-1}) = 2^p - 2,$$ if $2^{p-1}(2^p - 1)$ is an even perfect number, $D(x) = 2x - \sigma(x)$ is the ...
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1answer
42 views

Manufacturing desired planks from an existing pile of planks

Suppose there is a pile of commensurable planks that only may differ in lengths $0<a_1\leq\cdots\leq a_m$, which are to be used to manufacture planks of length: $0<b_1\leq\cdots\leq b_n$. ...
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1answer
168 views

Is there are similar conjecture like this??

Talking with my friend, my friend suggest impressive conjecture that For $i\in\mathbb{N}$, there are always exist natural number $r$ that satisfies $$\sum_{n=1}^{i} \frac{1}{n^r}=\frac{p}{q} , \...
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1answer
37 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 ...
3
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1answer
123 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 ...
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1answer
97 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 ...
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0answers
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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 ...
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1answer
68 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.
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0answers
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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\{...
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0answers
71 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 ...
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4answers
90 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.
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2answers
65 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 ...
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3answers
74 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)...
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4answers
115 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 ...
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0answers
102 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 ...
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0answers
31 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 $...
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11answers
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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,...
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1answer
118 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-...
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1answer
53 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 $\...
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1answer
77 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 ...
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1answer
44 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 ...
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0answers
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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 ...
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2answers
200 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 ...
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1answer
33 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 ...
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1answer
60 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 ...
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0answers
62 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 ...
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2answers
446 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
63 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
238 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
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1answer
124 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 ...
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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 ...