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        Questions tagged [prime-numbers]

        Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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        78 views

        On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$

        Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral. Is it true that $$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
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        Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$

        For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$ with $T_1=1$, where $p_k$ denotes the $k-th$ prime. So multiplying by $(-1)^n$ and telecoping gives that for ...
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        Prime numbers and sieving up to $q(x)=\log(x)(1+o(1))$

        Let $x,z\in\mathbb{R}_{+}$ Let $P_z = \displaystyle{\small \prod_{\substack{p \leq z \\ \text{p prime}}} {\normalsize p}}$ Let $I(x, z)=\{k \in \mathbb{N} \, | \, k \leq x \text{ and } \gcd(k, P_z)=...
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        202 views

        Complexity of representations of sets using elementary functions

        Fermat conjectured that $2^{2^n}+1$ is prime for every $n \in \mathbb{N}.$ Before even knowing about Euler's counterexample (that $2^{32}+1 = 641 \cdot 6700417$), you could possibly say that Fermat ...
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        From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes

        In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
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        28 views

        On values of $n\geq 1$ satisfying that for all primorial $N_k$ less than $n$ the difference $n-N_k$ is a prime number

        In this post I present a similar question that shows section A19 of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from ...
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        72 views

        Is the ratio of a number to the variance of its divisors injective?

        The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
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        On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question

        This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
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        2answers
        144 views

        On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$

        Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above. Is $I_s$ known to be convergent for any real number $s<1$ ?
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        59 views

        Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?

        I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
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        381 views

        On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity

        Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
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        Is there a Kolmogorov complexity proof of the prime number theorem?

        Lance Fortnow uses Kolmorogov complexity to prove an Almost Prime Number Theorem (https://lance.fortnow.com/papers/files/kaikoura.pdf, after theorem $2.1$): the $i$th prime is at most $i(\log i)^2$. ...
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        44 views

        What about series involving strong primes?

        I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
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        316 views

        A conjectural formula for the class number of the field $\mathbb Q(\sqrt{-p})$ with $p\equiv3\pmod8$

        Question. Is my following conjecture new? How to prove it? Conjecture. Let $p>3$ be a prime with $p\equiv3\pmod 8$, and let $h(-p)$ denote the class number of the imaginary quadratic field $\...
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        65 views

        An attempt to get a variant of Agoh–Giuga conjecture

        The idea of this post is an attempt to explore a variant of the so-called Agoh–Giuga conjecture. In past days, and today, I tried to think about variants of this conjecture exploring congruences about ...

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