Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

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7
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1answer
172 views

Conjecture: All $x^2+1$ primes for $x>90$ can be represented as the sum of five $x^2+1$ primes.

For all $x>90$, I assert that any $x^2+1$ prime may be written as the sum of five smaller $x^2+1$ primes. In fact, above that bound, I think a stronger conjecture holds where one of said primes can ...
4
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0answers
129 views

$2$'nd Law of thermodynamics in a relativistic gas?

Question I'm trying to understand in the context of a gas if a particular boundary condition can be used be to be the source of the $2$'nd law of thermodynamics: "Given an isolated system the entropy ...
6
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1answer
80 views

$C+ \frac 1 n C=[0,1+1/n]$ conjecture

Let $n$ be a positive integer. Let $C \subseteq[0,1]$ be the ternary Cantor set. I have made the conjecture $C+\frac 1 nC=[0,1+1/n]$ and am wondering if it is true or not. It is obviously true when $...
1
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0answers
70 views

A pre-successor hyperoperation: subplus

I wrote up these notes about a month ago and just found them. I could really use a second opinion or two. I notice there's a whole question about how my whole thesis is wrong, but that doesn't ...
6
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1answer
78 views

The crosshatch conjecture, on primes in $(p,p^2)$

If the first $p^2$ integers are laid out in a $p\times p$ square, every row and column will have at least one prime. Easily visualized as so: I recognize this should maybe be packaged as two ...
6
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1answer
81 views

$p$, $q$ and $\sqrt[n]{p} + \sqrt[n]{q}$ are rational, with the latter being non-zero. Are $\sqrt[n]{p}$ and $\sqrt[n]{q}$ rational?

Let $p, q \in \mathbb Q$, $n \in \mathbb Z^+$ and label $a = \sqrt[n]p, b=\sqrt[n]q$. Conjecture: If $a + b$ is a non-zero rational, then both $a$ and $b$ are rational. (Preliminary question: is ...
2
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0answers
23 views

Justification for the conjectures on self-avoiding walks

There are two conjectures on self-avoiding walks that I haven't seen any justification of. One is the asymptotic growth of the number of length $n$ self-avoiding walks, the form I've seen most often ...
6
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2answers
143 views

Conjecture: smallest missing mod value always yields previous prime

I've come up with a conjecture that seems similar in strength to Legendre's or Oppermann's, but maybe subtly different. Let $a_n$ be the smallest nonnegative value such that there is no $m$ in $1<...
1
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1answer
21 views

If a case of the Bunyakovsky conjecture holds for degree-$n$ $p(x)$, does it hold for all degree-$n$ $f(x)$ satisfying the criterion?

The Bunyakovsky conjecture states that if a polynomial $f(x)$ satisfies: The leading coefficient is positive The polynomial is irreducible over $\mathbb{Z}$ $f(1), f(2) \dots$ share no common factor (...
2
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0answers
166 views

Lower bound for the $n$-th record gap between primes in an arithmetic progression

(This is a natural counterpart to question 3132001 which deals with upper bounds.) Let $q$ and $r$ be coprime integers, $1\le r < q$, and consider the arithmetic progression $$ r, \ r+q, \ r+2q, \ ...
0
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1answer
61 views

Does “smooth finite average” exist?

(That question was answered. I realized I asked a wrong question. So here is a modified question.) I will call an average any continuous function $f(v_1,\dots,v_n)$ of $n$ arguments such that it lies ...
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2answers
66 views

Does “finite average” exist?

I will call an average any continuous function $f(v_1,\dots,v_n)$ of $n$ arguments such that it lies in the closed interval $[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$, is symmetric for all ...
0
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0answers
33 views

Conjecture about generalized means

Conjecture Every continuous function $f(v_1,\dots,v_n)$ of $n$ arguments such that it lies in the closed interval $[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$, is symmetric for all ...
2
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1answer
41 views

A possible disproof for the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

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 $D(z):=2z-\sigma(z)$, and the sum of the aliquot divisors by $s(z):=...
7
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2answers
89 views

For all $n\in\mathbb{N}$ with $n>1$ there exists an $m\in\mathbb{N}$ with $m<n$ such that $mn+1$ is prime.

I was using Python to explore some interesting lexicographically earliest sequences. I was looking into the sequence of numbers such that $a(1)=1$ and $a(n)$ is the smallest number such that $a(k)n+1$ ...
-4
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1answer
125 views

Why is that a conjecture is written and its proof can not be done for years? [closed]

Why is it that if a conjecture is written and its proof can not be done for years, means if its statement is understandable in language, why is the proof not possible if the conjecture is correct? E.g....
2
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0answers
54 views

If $q^k n^2$ is an odd perfect number with special prime $q$, then can $n^2 - q^k$ be a cube?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$. This question is an offshoot of the following earlier post: If $q^k n^2$ is an odd perfect ...
5
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1answer
184 views

Has this basic number theory conjecture been discovered or proved before?

So I was sitting in a car bored for two hours playing around with the calculator on my phone, and I discovered that by picking two numbers (at least one being odd and both being not divisible by each ...
1
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1answer
70 views

A conjecture about the number of divisors of a natural number

Conjecture: $\tau(n)\mid\tau(n^2)\iff$ $n$ is a perfect square and $\sqrt n=p^2s$, where $p$ is prime and $s$ is a non prime squarefree number such that $\gcd(p,s)=1$. $\tau(n)$ is the ...
2
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1answer
91 views

Does this procedure always generates at least one prime?

Suppose that $p_n$ is the $n$-th prime and $n \neq 1,2$. To every $p_n$ we can associate $(n-2)$-tuple $(2p_n+p_{n-1},...,2p_n+p_2)$ and from some calculations that I have done it seems that it ...
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0answers
66 views

Prime twins $ (3^n - 2, 3^n - 4) $ conjecture

Let $n$ be a positive integer. Conjecture There are infinitely many prime twins of the form $$ ( 3^n - 2, 3^n - 4) $$ Examples include $$(3^2 - 2,3^2 - 4) = ( 7,5 ) $$ $$ ( 3^{37} - 2 , 3^{37} - ...
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0answers
73 views

Every number is a sum of (exactly/at most) some (distinct or not) primes

Introduction I have observed some heuristic results, and also attempted to formulate a proof, of the conjecture I formulated based on those heuristics. But first, I want to ask about relevant ...
7
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4answers
400 views

Some lesser known open problems/ conjectures in number theory

What are some lesser known problems/ conjectures in number theory ( especially on prime numbers ) which have evolved in these recent years and didn't got much of attention, and obviously wasn't ...
2
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1answer
105 views

Are there any example of conjectures which have been disproved, causing other maths built on it, to be wrong?

I am interested in the consequences of putting faith in conjectures which have not yet been proved beyond all doubt. Has there ever been important conjectures which when disproved have led to the ...
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0answers
56 views

Regarding the Erdos-Szemeredi Sum/Product Conjecture

Edit: The theorem that was proved is here: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szemer%C3%A9di_theorem , basically a statement that either there can be many unique elements in a sum matrix ...
7
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0answers
165 views

A generalization of the abc-conjecture?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
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1answer
87 views

Yet a conjecture about the prime counting function.

This is a conjecture emanating from random tests with my set routines written in Forth. $a>b \;\wedge \pi(a+b)=\pi(a)+\pi(b)\implies b<11$ Furthermore, if $b\in\mathbb N$, then $b\in\{0,...
2
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1answer
65 views

Equivalency of Grimm's Conjecture to Legendre's conjecture

Grimms conjecture If $n + 1$, $n + 2$, …, $n + k$ are all composite numbers, then there are k distinct primes $p_ᵢ$ such that $p_ᵢ$ divides $n + i$ for $1 \leq i \leq k$. For example, for the range $...
-2
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2answers
148 views

Have I just proven the Twin Prime Conjecture? [closed]

For those who aren't familiar with it, the Twin Prime Conjecture wonders if there are an infinite number of prime coordinate pairs $(p, q)$ such that $p=q+2$. I'm wondering if I've proven the Twin ...
7
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1answer
102 views

Can this heuristic argument be useful to prove: Sum of digits of $a^b$ equals $ab$ conjecture?

I've been trying to write a proof for the following conjecture (from this question): Let $s\left(a^{b}\right)$ denote the sum of the digits of $a^{b}$ in base $10$. Then the only integer values $a$,$...
13
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0answers
151 views

Rare gaps between integers having the same number of co-primes

Let $\varphi(x)$ be the Euler totient function. For $k = 1,2,3,\ldots$ I calculated the number of solutions of $\varphi(x) = \varphi(x+k)$. I observed that we have very few solutions when $k = 3,9,15,...
2
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2answers
84 views

A pattern on finding squares for which their sum is $\Big(\sum\limits_{i=0}^j x^i\Big)^2$

Apologies for the inactivity. I haven't been doing so well in life, lately, but I'm glad to be back at some maths! Here's a pattern I discovered. I'm not good at explaining with words, so hope you get ...
7
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0answers
119 views

Conjecture: There is no $n\neq3$ such that $\arctan(1)+\arctan(2)+\cdots+\arctan(n)=\frac{k\pi}{2}$, with $k$ integer.

Fact: $\arctan (1) + \arctan (2) + \arctan (3) = \pi$. Conjecture: there is no $n \neq 3$ such that $\arctan (1) + \arctan (2) + \cdots + \arctan (n) = \frac{k\pi}{2}$, with $k$ integer. I know a ...
1
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2answers
60 views

Is there a conjecture suggesting if some other conjecture is true for all $x<n$, then it's true for all $x$?

So everybody here probably knows if numerical testings support a conjecture, that conjecture isn't necessarily true for larger numbers. In fact,there have been many times where a conjecture was proved ...
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0answers
87 views

$\int_{0}^{\infty}\sin({x f({x})}) d x$ converges for positive monotone increasing unbounded continuous $f({x})$

Let a continuous speed function $f:\mathbb{R}^+\to\mathbb{R}^+$ exist such that $ [{\{a,b\}\subseteq\mathbb{R}^+\land a<b}] \Rightarrow f(a)<f(b) $. Then $\int_{0}^{\infty}\sin({x f({x})}) d x$ ...
1
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1answer
173 views

Is there a Collatz 'branch' stopping time equation?

Definitions: -Collatz: odd: $\frac{3x+1}{2}$ | even: $\frac{x}{2}$ -Branch: Starting at an odd number and increasing until reaching an even number. Then decreasing until reaching another odd number (...
1
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2answers
37 views

Manipulating sets into periodic sets

Consider the set $[0,1,1,0,1,0,1,0,0,0,1,...]$. The nth element in this set is equal to $1$ if n is prime and is else equal to $0$. Here are my question: Is this Set non-periodic (If you write it as ...
0
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2answers
48 views

Conjecture: $n\in\mathbb{Z}^+\Rightarrow 0=\int_{0}^\infty x^{4n-1}\sin(x)e^{-x}dx$

Conjecture: $n\in\mathbb{Z}^+\Rightarrow 0=\int_{0}^\infty x^{4n-1}\sin(x)e^{-x}dx$ This has been verified with WolframCloud for $1\leq n\leq 500$.
0
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1answer
59 views

Natural numbers that are square sums of four different nonzero integers

It's known that any natural number can be expressed as a sum of four squares. Conjecture: A natural number can be written as a square sum of four different nonzero integers if and only if it not ...
3
votes
1answer
79 views

Conjecture: Any sufficiently big sum of three squares can be written as a square sum of three different natural numbers greater than zero

A natural number can be written as a sum of three squares if and only if it's not of the form $4^m(8n+7)$ for natural numbers $m,n$. I'm curious about such numbers that can be written as the square ...
4
votes
1answer
117 views

$n^{th}$ Dimensional Prime Nexus Conjecture (Hamilton's Path & Cycle)

The system and proposition of Prime Nexus - We have to arrange $n>1$ Distinct natural numbers in a sequence such that the adjacent elements sums upto any Prime number. Convenience is when we set ...
0
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2answers
170 views

Why problem with simple formulation is so hard?

If you ever heard about Collatz conjecture, you know that it is understandable even for middle school students, but no one has solved it yet. The problem is to prove or to disprove that starting with ...
2
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0answers
51 views

Determinant Optimization Conjecture

Let the maximum value of $n \times n$ determinant with $n \in\mathbb{N} >1$ be $D_n$ . If each of the elements in determinant is either $+1$ or $-1$ , Then the successive values of $D_n$ makes a ...
4
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0answers
91 views

a version of the famous $n!+1=m^2$ problem

It is still unknown if $n!+1=m^2$ has only 3 solutions, i.e. for $n=4,5,7$. $n!+1$ being a perfect square https://en.wikipedia.org/wiki/Brocard%27s_problem What about my problem: Are there ...
2
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0answers
101 views

Conjecture on Prime Numbers and Exponential Equipment

How to generate a rigorous proof for - The specific conjecture which states that we can always frame every Prime number in an exponentially framed equation with perfect cubes & squares. I'll ...
2
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0answers
74 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts ...
10
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2answers
206 views

Conjecture $\sum_{n=0}^\infty a_n= \frac{1}{2}-\frac{7 \zeta(3)}{2 \pi^2}$

Working with some integrals I stumbled upon the following slowly converging series: $$ S = \sum_{n = 0}^{\infty}\left(-1\right)^{n} \left[n + \frac{3}{2} + \left(n + 1\right)\left(n + 2\right) \log\...
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0answers
36 views

If $q^k n^2$ is an odd perfect number with special prime $q$, then does $(q+1)/2 \mid \sigma(n^2)$ hold?

Let $\sigma(M)$ denote the sum of divisors of the positive integer $M$. If $N$ is odd and $\sigma(N)=2N$, then $N$ is called an odd perfect number. Euler showed that an odd perfect number, if one ...
0
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2answers
58 views

Problem understanding Legendre's conjecture

"Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime between $n^2$ and $(n + 1)^2$ for every positive integer $n$" (https://en.m.wikipedia.org/wiki/Legendre%...
2
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2answers
121 views

An iteration formula I found (please don't jump at me if it's already been discovered)

$\sin^2 \theta$"> Where the modulus-like symbol actually denotes iteration of a radical function. Sorry for the messy work everyone- I am new to this stuff, and I literally just found this iteration ...

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