# 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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### Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
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### A conjecture about binary palindromes and arithmetic derivatives

Corrected question. From the sequence of binary palindromes A006995 (eg. 1001001001001) the sequence of possible gaps between consecutive palindromes contain the elements: ...
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### Product of Consecutive integers problem [duplicate]

Product of Consecutive integers problem I've begun pondering the following problem, and have found myself unable to advance on it; it goes like this: Prove that the product of three consecutive ...
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### Beal's Conjecture can the following specific case be proved?

According to wikipedia's page on Beal's Conjecture, it states: $A^x + B^y = C^z$, where $A, B, C, x, y$ and $z$ are positive integers and $x, y$ and $z$ are all greater than $2$, then $A, B$ and $C$ ...
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### A problem with the equivalent form of Frankl conjecture.

The Union Closed Sets Conjecture (or Frankl conjecture) is described in this link: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture. It is known that this problem has an equivalent form in ...
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### On Andrica's conjecture.

We have Andirca's conjecture as $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$. I have no clue if this is a proof. I was just messing around. (And plus, my methods are probably too elementary). Denote $g(n)$ as ...
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### Integer solutions to $r\cdot a^2+t=s\cdot b^2$ forms sequences with $a_{n+1}\cdot b_n-a_n\cdot b_{n+1}$ is invariant

Conjecture: Given natural numbers $r,s,t$. Suppose there are an infinite number of solutions $a_n,b_n \in \mathbb N$ to $ra_n^2+t=sb_n^2$, where $a_n, b_n$ correspond to increasing sequences of ...
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### Conjecture regarding the sum of prime factors of $x!$ and $(x-1)!$?

I think using dodgy means I can show the following: Let $\lambda(x)$ be the sum of primes in $x$. For example: $$\lambda(2) = 2$$ $$\lambda(4) = 2 + 2 = 4$$ $$\lambda(6) = 3 + 2 = 5$$ Then for ...
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### Does the Bunyakovsky conjecture apply for every cyclotomic polynomial?

The Bunyakovsky conjecture states that for every irreducible polynomial $\ f\in \mathbb Z[x]\$ with positive leading coefficient, there are infinite many primes of the form $\ f(m)\$ , where $\ m\$ ...
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### Can you prove this number theory conjecture?

I think I can prove the following (using a conjecture): $$2n (\sum_{i=2}^n p_i ) \geq p_n(p_n+1)/2 - 6$$ where $p_i$ is the $i$th prime. Can one prove this without any conjecture?
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### Conjecture : A lower bound for the prime counting function .

Well it take my a little bit of time to find it but now I think that my conjecture is ready . Conjecture : Let $n\geq 100$ and then define the sum : S(n)=\sum_{k=1}^{n}\frac{1}{\operatorname{...
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### Is there always a prime $p$ which falls short of $2^n$ by at most a square-root factor?

For fixed $n > 0$, does there always exist a prime $p$ such that $p < 2^n \leq \lceil p + \sqrt{p} \rceil$? Would this follow from any known conjectures? This follows immediately from Opperman's ...
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### Bézout giving some spare coins to Frobenius? (the Coin Problem)

Let $a$ and $b$ be relatively prime positive integers and write the two minimal Bézout's identity pairs, $\tag 1 (s)\,a + (t)\,b = 1$ $\tag 2 (u)\,a + \;(v)\,b = 1$ where the extended Euclidean ...
A Jordan-Polya number A001013 is a product of factorials $\prod^m_{k=1}a_k!$. For $n=2,\dots,12$ there are Jordan-Polya numbers $a,b$ such that $n!=a+b$. ...