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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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        Additive and multiplicative convolution deeply related in modular forms

        From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
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        63 views

        Primes to base 30: n*30+{1,7,11,13,17,19,23} or n*30+{7,11,13,17,19,23,29} not possible to be prime for all values in {}, why? [closed]

        Using base 30 I just checked it for the first 2^32 numbers and it seems there is no such sequence that n*30 + 1, n*30+7, n*30+11, n*30+13, n*30+17, n*30+19, n*30+23 are all primes for some n. Another ...
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        1answer
        116 views

        On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function

        Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime. Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
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        115 views

        Is the sequence $\left(\pi\left(\frac{n(n+1)}2+1\right)\right)_{n\ge1}$ an addition chain?

        A (finite or infinite) strictly increasing sequence with the initial term $1$ is called an addition chain if each term after the initial one can be written as the sum of two earlier (not necessarily ...
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        52 views

        Is the fundamental partition associated to $n$ the partition of $r_{0}(n)$ in $k_{0}(n)$ parts that maximizes entropy?

        As usual, under Goldbach's conjecture, let's define for a large enough composite integer $n$ the quantities $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$ and $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-...
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        1answer
        205 views

        Why does each integer between two consecutive primes have at least one “unique” non-trivial divisor?

        Why does each integer $x$ between two consecutive primes have at least one non-trivial divisor that unique on set of all integers between these two consequtive primes except $x$? We call a divisor $d$...
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        275 views

        Iterated Twin Prime conjecture [closed]

        "Conjecture. The sum of a twin prime pair greater than or equal to 24 can be expressed as the sum of two twin prime pairs." Examples: ...
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        169 views

        Prime character sums

        Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
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        4answers
        961 views

        Can anything deep be said uniformly about conjectures like Goldbach's?

        This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our ...
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        2answers
        262 views

        Least (and largest) possible number of non-relatively prime pairs among consecutive integers

        Given a set of positive integers consider the graph whose vertices are those integers, two of which are joined by an edge if and only if they have a common divisor greater than 1 (i.e, they are not ...
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        126 views

        Consecutive integers each of which has a large prime factor

        There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor? More precisely, let $P(n)$ be the ...
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        votes
        1answer
        100 views

        Cipolla's Prime numbers function: Computing the coefficients of the polynomial

        In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited: [1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti ...
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        1answer
        228 views

        Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd?

        Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not ...
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        191 views

        On Primes in Arithmetic Progressions

        I was wondering if the following approach is being attempted to prove the twin-prime conjecture. Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
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        2answers
        211 views

        How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?

        I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that $$ \prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...

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