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Results tagged with prime-numbers
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user 10035
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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Extremal sets generating primes
Given $n_1,n_2,m>0$ is there always a $t<\infty$ such that there are odd $a_1,\dots,a_{n_1}\in\{1,\dots,t\}$ and even $b_1,\dots,b_{n_2}\in\{1,\dots,t\}$ with each $a_i^2+b_j^2$ a distinct prime $>m$? …
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On a definition of sensitivity of primes in base-$2$
Given an odd integer in $\mathbb Z_{\geq0}$ of $n$ bits let $a_{n-1}a_{n-2}\dots a_1a_0$ be its binary expansion where $a_{n-1}=a_0=1$.
Call an $n$ bit prime $f(n)$-sensitive (similar to sensitivity o …
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Provably undecidable problems within prime numbers context
A colleague of mine was stating there are no known undecidable statements that have explicit connection with prime numbers. What does this mean? I understand that it is unknown whether Goldbach conjec …
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Expected number of primes of particular size and from a linear form
Given two distinct primes $P_1$ and $P_2$ picked randomly and uniformly in the interval $[T^2,2T^2]$ consider the set $\chi(P_1,P_2)$ of numbers of form $$xP_1-yP_2$$ where $x,y$ are in $[0,T^{1+\epsi …
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Small solutions to modular hyperbola
Fix $\ell\in\Bbb N_{>1}$ and small $0<\epsilon\ll1$.
Given $r_1,\dots,r_\ell\in(0,1)$ with $\sum_{i=1}^\ell r_i=1$, is it possible to always find $\Omega(p^{\ell\epsilon'})$ solutions $x_i\in\Bbb …
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Density of triple primes
The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.
Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds.
Is there any reason t …
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On complexity of a particular prime problem
Is the following problem in $PH$ and is it complete for any class?
Problem: Is the $i$th bit of the $m$th prime $1$?
It appears to require a counting quantifier which has to demonstrate witness is the …
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On the Carmichael Lambda function
Let Carmichael function be denoted by $\lambda(n)$.
Consider the set $I_m=\{n:\lambda(n)=m\}$.
What is known about the cardinality of $I_m$?
Let $P_m=\{p\in Primes: p|\ell \mbox{ for some }\ell\in I …
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Asymptotics of Product of consecutive primes
I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
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Primes which are safe and Sophie Germain
If $p$ is a Sophie Germain prime then $2p+1$ is safe prime.
If $2p+1$ is safe prime then $p$ is Sophie Germain prime.
What is their conjectured distribution of primes $p$ which are both Sophie Germai …
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On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain...
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have …
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Negative Dirichlet Pigeonhole Principle [closed]
From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where $t …
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counting irreducible factors
In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer.
Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ …
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Effective prime number theorem
The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at most …
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On certain number theoretic sextuples?
Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,c,d,e,f$ with $a,b …
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