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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.
3
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1
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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 …
- 13.9k
0
votes
0
answers
58
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Factoring totient of a prime
Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem?
What about when $2p+1$ is also a prime?
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2
votes
0
answers
68
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Twin prime distribution centering twice a semiprime
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?
- 13.9k
1
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0
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63
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Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 13.9k
2
votes
1
answer
536
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Is there a Cramer's conjecture for Sophie Germain primes?
A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie Germ …
- 13.9k
1
vote
1
answer
306
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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 …
- 13.9k
3
votes
0
answers
115
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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 …
- 13.9k
0
votes
0
answers
102
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On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four integ …
- 13.9k
1
vote
0
answers
95
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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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0
votes
0
answers
134
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On a deterministic primes search problem
I feel the following problem might be resolved already. But I could not find any related answers.
If $p_1,p_2,\dots,p_t$ are primes where $2\leq t=o(\log n)$ is there a prime within $$\prod_{i=1}^tp_ …
- 13.9k
1
vote
1
answer
110
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Succinct polynomial sized representation of balanced bipartite graphs whose perfect matching...
Is there a $P$ time definable sequence of succinct polynomial sized representation of balanced bipartite graphs whose number of perfect matchings is a primorial?
For factorial a complete bipartite gra …
- 13.9k
5
votes
0
answers
200
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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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0
votes
0
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62
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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 …
- 13.9k
1
vote
0
answers
90
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Smooth number pairs satisfying a congruence
Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\ell …
- 13.9k
0
votes
1
answer
248
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How differently would we model the distribution of primes if prime gap is larger?
Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime.
How differently would primes be modeled if gaps of $O …
- 13.9k