# Tagged Questions

Use this tag if your question is about a well-known conjecture or a conjecture of your own.

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### Twin prime conjecture implies $\limsup_{n\to\infty}\frac{\sigma(n)\pi(n)}{n^2}\left(\pi(\log n)-\frac{\pi_2(\log n)}{2C_2}\right)=e^{\gamma}$?

Let $\sigma(n)$ the sum of positive divisor function, $\pi(x)$ is the prime counting function, $\pi_2(x)$ is the twin prime counting function (we will assume that Twin prime conjecture holds), $C_2$ ...
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### Is it true/known/important that $(\log p_n)/n$ is nonincreasing, where $p_n$ is the $n$th odd prime number?

First thing first, I would like to apologize in advance for my poor knowledge of Maths and English. I'm an Italian student and after asking to all the mathematicians and Maths teachers in my town, I ...
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### I conjecture that there are infinitely many linear relations $p_n + p_{n + 3} = 2 p_{n + 2}$ in the sequence of primes!

Let $p_1 < p_2 < p_3 < \ldots < p_n < \ldots$ be the sequence of primes (with $p_1 := 2$ as usual). Conjecture: There are infinitely many positive integers $n$ for which \begin{...
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### a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
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### conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
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### Can the sum of 3 unique primes be expressed as the sum of 2 primes?

Let's consider the example, $$3 + 11 + 19 = 33 \\ 2 + 31 = 33$$ we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case ...
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### If $(a_n)$ is increasing and $a_n^{1/c^n}\to\infty$, then $\sum\frac1{a_n}$ is irrational?

I $\DeclareMathOperator\lcm{lcm}$am trying to generalise the result from this question: If $(a_n)$ is increasing and $\lim_{n \to \infty} a_n^{1/2^n} = \infty$, show that $\sum_{k=1}^{\infty} 1/{a_n}$ ...
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### Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$... Bertand's postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this? any ideas?
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### Sum involving the number of zeros of $k$

I'm currently interested in sums involving digit-functions. Especially, I'd like to calculate the following sum: $$s=\sum_{k=1}^{\infty}{\frac{a_0(k)}{k\left(k+1\right)}}$$ Where $a_0(k)$ is the ...
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### A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: \begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ &+720\ln^32\cdot\ln3+360\ln2\cdot\ln^...
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### The asymptotic behaviour of triples $n!+q^{n!}=c$, where $q$ is the first prime greater than $n$, and abc conjecture

For a large positive integer $n$, let $q=q(n)$ (below we denote this $q=q(n)$ by $q_{N}$ because we assumed that is the $N$-th prime number) the first prime number which is (strictly) greater that $n$....
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### Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...