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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.
29
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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 $i(\log i)^2$. T …
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6
votes
1
answer
653
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On permuted sum of squares of primes in a list
We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, $\sigma_1(\cdot)$,$\sigma_2(\cdot),\dots,\sigma_t(\c …
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5
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Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?
Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.
Is there a version of …
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5
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3
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3k
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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?
- 13.9k
4
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0
answers
407
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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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4
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0
answers
211
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What is the complexity class of this problem without Cramer's conjecture?
The problem 'Given $0<a<b$ is there a prime in the interval $[a,b]$?' is in $\mathsf{NP}$. If we assume Cramer's conjecture the problem is in $\mathsf{P}$ since if $b-a>(\log a)^{2+\epsilon}$ at any f …
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4
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0
answers
279
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Analog of Euler's factoring technique
Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Euler's two squares factoring states that numbers …
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4
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answer
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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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3
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275
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Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is mag …
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3
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1
answer
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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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3
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0
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130
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Improving prime number generation probability?
Deterministic generation of primes in polynomial time is unknown.
Is there a way to probablistically in $O(n^c)$ time bound for some $c>0$ generate polynomially $\Omega(n^c)$ many integers in $[0,2^ …
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3
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0
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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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3
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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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3
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answer
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PNT analog for primes inside a structured set
Let $\Bbb T$ be the set of all square free integers with ordering derived from $\Bbb N$. Essentially $PNT$ says if you pick $\log N$ integers less than $N$ you can expect one of them to be prime.
W …
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3
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2
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331
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On generating squarefree integers and primes?
Given an $\alpha\in(0,1)$ and $n\in\Bbb N$ what are some known deterministic algorithms to sample $O(n^\alpha)$ (not just get one) square free integers of $n$ bits? Is it $O(n^{\alpha})$ complexity?
…
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