# 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: prime sum of two squares between every pair of consecutive squares

It appears that between every $n^2$ and $(n+1)^2$, for $n \geq 1$, there's at least one prime that is a Pythagorean prime and can be represented as the sum of two squares. In fact, it turns out that ...
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### Is there any mathematical conjecture that is successfully applied in the real world in spite of being yet unproven?

I'm a philosophy student and I'm writing a thesis that makes a few comparisons between ethics and mathematics. My knowledge of mathematics is limited, however, and in the process of making my ...
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### Prove a conjecture, balls in boxes, n steps

My uncle gave me the following problem to work on (just for fun), he doesn't know whether the problem has a solution. I haven't been able to solve it and I give up, I don't think my current knowledge ...
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### Grimm's Weak Conjecture

On Wikipedia it says that the following weaker version of Grimm's conjecture is also still open: "A product of $k$ consecutive composite numbers has $k$ pairwise distinct prime factors." Is it true, ...
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### Question related to Sophie Germain Primes

Assuming $p_i$ is a prime consider the sequence defined by $q_1=p_i$ and $q_n=2\ q_{n-1}+1$ for $n>1$. Let $d$ be the greatest value $n$ for which $q_1..q_n$ are all Sophie Germain primes which I ...
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### Conjecture: There are infinitely many $N \in \Bbb{N}$ such that $p$ a prime $p \leq \sqrt{N+1} \implies p \mid N$?

Conjecture. There are infinitely many $N$ such that if $p$ is a prime $\leq \sqrt{N+1}$ then $p \mid N$. Is this another hard to prove number theory conjecture, or do you have some idea of how to ...
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### Conjecture on the distribution of prime numbers and the prime counting function

I conjectured this result and i want to put it to the test. Let a perfect sequence be an arithmetic progression $(a_i)_{i=1}^{\infty}$ such that the sequence $(\pi(a_i))_{i=1}^{\infty}$ is also an ...
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### A pattern with third coefficients of sums of powers.

So I have heard countless times of Bernoulli Numbers and its relation to the sums of powers. Inspired by this (as well as Chebyshev polynomials), I decided to look at the sums of powers myself and ...
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### Is the following an existing conjecture or a conjecture at all?

the following floated to my mind today, can you verify if it stands to be true, or is a pre-existing conjecture. If not, can you correct me? And if it is one, can you prove it? Prime factorisation of ...
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### Conjecture: every set of $n$ equally spaced integers $\leq n^2$ contains a prime.

My claim: Any set of exactly $n$ integers in the range $[1,n^2]$ which are an arithmetic sequence and share no common factor will contain at least one prime. Let $n\geq 2$ be a natural number, and ...
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### Bass-Quillen conjecture for non-affine case

Bass-Quillen conjecture expects that any vector bundle on $U\times \mathbb{A}^1$, to be extended from $U$. Here $U$ is a regular affine scheme. Being affine is an essential part of the conjecture, you ...
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### On proving an infinite-nested radical

I was playing around with square roots and I noticed that the number $1$ can be seemingly expressed as an infinite nested radical with an easy pattern. I then noticed that if this is true, this would ...
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### Does the fraction of distinct substrings in prefixes of the Thue–Morse sequence of length $2^n$ tend to $73/96$?

Recall that the Thue–Morse sequence$^{}$$\!^{}$$\!^{}$ is an infinite binary sequence that begins with $\,t_0 = 0,$ and whose each prefix $p_n$ of length $2^n$ is immediately followed by its ...
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### list of open conjectures which are solved by students in high-Middle school level or receprocally?

I know only Snevily's conjecture which it was proved in 2009 by Bodan Arsovski. and He was a high-school student at the time., Are there other conjectures which are solved by students in high ...
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### If $q^k n^2$ is an odd perfect number with special prime $q$, then $k \neq 1$ (MSE version)

Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
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### How were the five large solutions to the Fermat-Catalan conjecture found?

The Fermat-Catalan conjecture is the statement that the equation$a^m + b^n = c^k$has only finitely many solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers ...
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### Attempt to get a characterization of even perfect numbers from an equation involving the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(n)$ for integers $n\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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