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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.
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 …
- 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 …
- 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
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
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \ …
- 19.6k
18
votes
1
answer
649
views
How hard is it to find a prime number with given primitive roots?
Assume that we randomly choose a 100-digit prime number $p$,
record which of the first 1000 prime numbers are primitive roots
modulo $p$, and then forget about $p$. —
How easy or how difficult is it t …
- 19.6k
18
votes
2
answers
2k
views
Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers
To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes
infinitely many prime numbers as values. Is it known so far whether there is at all any
polynomial $P \in \math …
- 19.6k
15
votes
Accepted
Lower density of {primes} times themselves
There is no such $m_0$, due to the Erdős–Kac theorem.
- 19.6k
14
votes
How did Cole factor $2^{67}-1$ in 1903?
While the question how Cole factored $N := 2^{67}−1$ has already been answered by quid,
David Speyer's more general question is how someone could find the factors of that
number in $100000$ minutes of …
- 19.6k
10
votes
Accepted
Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?
Carl Pomerance conjectured in
On the Distribution of Pseudoprimes, Math. Comput. 37, 587-593 (1981)
that for large $x$, the number of pseudoprimes $\leq x$ is
$$
\frac{x}{e^{(1+o(1))\log{x}\frac{\ …
- 19.6k
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. …
- 19.6k
9
votes
Probability that randomly chosen integers from a restricted set of natural numbers are coprime
First of all, note that there is no canonical notion of equidistribution on a countable set
like the integers.
When asking for the probability that $k$ 'randomly chosen' integers are coprime,
it is m …
- 19.6k
8
votes
Accepted
Is it true that $p^2+1$ is square free if $p>7$ is a Mersenne prime
No, this is not true. -- For example $p := 2^{2203}-1$ is a Mersenne prime
(cf. http://en.wikipedia.org/wiki/Mersenne_prime), but $p^2+1$ is divisible by $5^2 = 25$.
Edit: To answer D. Burde's questi …
- 19.6k
7
votes
Is the $n$-th prime $p_n$ expressible as the difference of coprime $A, B$ such that the set ...
Barry Cipra has already computed the first few values.
The next couple of numbers $p_n$ are
$13 = 5 \cdot 11 - 2 \cdot 3 \cdot 7$,
$17 = 2 \cdot 7 \cdot 13 - 3 \cdot 5 \cdot 11$,
$19 = 2^2 \cdot 3 …
- 19.6k
6
votes
1
answer
378
views
Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)
Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)} = m-1$ …
- 19.6k
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