# 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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### 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 ...
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### $2$'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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### $\int_{0}^{\infty}\sin({x f({x})}) d x$ 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 $\int_{0}^{\infty}\sin({x f({x})}) d x$ ...
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### 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 (...
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### 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 ...
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### 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 $1\leq n\leq 500$.
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### 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 ...
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### 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 ...
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### $n^{th}$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $π(x+y) - π(x) ≤ c·y/\ln(y)$ 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 ...
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### 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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### 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 ...
"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%...
$\sin^2 \theta$"> 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 ...