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Results tagged with prime-numbers
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user 28104
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.
1
vote
0
answers
71
views
Is there an upper bound on the number of partitions of a finite set of primes into 3 sets th...
Is there an upper bound on the number of partitions of a finite set $S$
of prime numbers into 3 sets $A$, $B$ and $C$ for which the following holds?:
$$
\prod_{p \in A} p \ + \ \prod_{p \in B} p \ = …
- 19.6k
39
votes
1
answer
2k
views
Prime number races in 2 dimensions
Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, I t …
- 2,821
18
votes
1
answer
649
views
How hard is it to find a prime number with given primitive roots?
Assume that we randomly choose a 100-digit prime number $p$,
record which of the first 1000 prime numbers are primitive roots
modulo $p$, and then forget about $p$. —
How easy or how difficult is it t …
- 186
18
votes
2
answers
2k
views
Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers
To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes
infinitely many prime numbers as values. Is it known so far whether there is at all any
polynomial $P \in \math …
CommunityBot
- 1
3
votes
0
answers
145
views
The bias of consecutive prime numbers towards being incongruent modulo 3
Given a positive integer $n$, let $f_1(n)$ denote the number of pairs of
consecutive prime numbers $\leq n$ which are incongruent modulo 3, and let
$f_2(n)$ denote the number of pairs of consecutive p …
- 19.6k
43
votes
1
answer
1k
views
Can't one walk to infinity on the prime numbers with finitely many distinct affine steps?
Let $(a_1,b_1), \dots, (a_k,b_k)$ be finitely many pairs of positive
integers, and let $\Gamma$ be the graph whose vertices are the prime
numbers and in which two vertices $p$ and $q$ are connected by …
CommunityBot
- 1
9
votes
2
answers
527
views
Sign of permutation induced by modular exponentiation
Given a prime number $p$ and a primitive root $a$ modulo $p$, let
$\sigma_{a,p}$ denote the permutation of the set $\{1, \dots, p-1\}$ which
maps $b$ to $a^b$ modulo $p$.
Question: Let $p$ be fixed. …
- 109k
5
votes
0
answers
176
views
Can the integers in an easily computable sequence free of prime numbers always be factored e...
Call a sequence $(a_n)$ of positive integers easily computable
if there is a constant $C$ and an algorithm which computes $a_n$ from
$n$, $a_1, \dots, a_{n-1}$ and a finite number of integer constant …
- 19.6k
2
votes
1
answer
202
views
Endomorphism of the symmetric group of the set of positive integers via action on the prime ...
For a positive integer $n$, let $p_n$ denote the $n$-th prime number.
Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$
be the monomorphism which maps a permutation $\sigma$ to …
- 34.3k
19
votes
1
answer
2k
views
How many primes can there be in a short interval?
Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \ …
- 114k
2
votes
Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$
Given a positive integer $k$, the natural density of the set of positive
integers $n$ whose largest prime factor is smaller than the $k$-th root of $n$
is estimated by the value of the Dickman functio …
- 19.6k
1
vote
Accepted
Set of triple-primes satisfying a certain equation
Yes, there is precisely one such triple of distinct primes
with $\alpha \leq 500$, namely $(3, 11, 31)$.
It would be a surprising coincidence if there are further
such triples for larger $\alpha$.
- 19.6k
10
votes
Accepted
Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?
Carl Pomerance conjectured in
On the Distribution of Pseudoprimes, Math. Comput. 37, 587-593 (1981)
that for large $x$, the number of pseudoprimes $\leq x$ is
$$
\frac{x}{e^{(1+o(1))\log{x}\frac{\ …
- 19.6k
38
votes
Accepted
Does the equation $241+2^{2s+1}=m^2$ have a solution?
To answer your first question: there is indeed no $s$ such that
$241+2^{2s+1}$ is a perfect square. -- Proof: $2^{2s+1}$ is always
congruent to either $2$, $8$ or $32$ modulo $63$, which makes
$241+2^ …
- 19.6k
15
votes
Accepted
Lower density of {primes} times themselves
There is no such $m_0$, due to the Erdős–Kac theorem.
- 19.6k