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Results tagged with prime-numbers
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user 4556
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.
11
votes
1
answer
430
views
First prime of the form $x_i$ for $x_0=658$ and $x_i=1+2x_{i-1}$
Given an initial integer $x_0>0$, one can consider the first prime of the recursive sequence $x_i=1+2x_{i-1}$.
Naïvely such a prime should exist for $x_0$ arbitrary since the sequence $\log(x_i)$ is a …
- 17.9k
0
votes
0
answers
101
views
Sequence $a_1,a_2,\ldots$ with $a_j\in\lbrace 1,2,\ldots,j\rbrace$ such that almost all $\su...
Are there prime-numbers having infinite left-expansions of non-zero coefficients in the factorial number system involving only prime numbers?
The question is really in the title : Is there an infinite …
- 17.9k
8
votes
1
answer
185
views
An abelian group associated to divisors of an integer $N$
A divisor $d$ of a natural integer $N$ defines a permutation
of $\{0,\ldots,N-1\}$ by considering
$$x\longmapsto \pi_{d\vert N}(x)=\left\lfloor \frac{x}{d}\right\rfloor+\frac{N}{d}
\left( x\pmod d\rig …
- 17.9k
0
votes
1
answer
156
views
A property related to permutations with coprime adjacent values
Sequence A76220 of OEIS enumerates (up to $n=25$) the number $a_n$
of permutations $\sigma$ of $\lbrace 1,\ldots,n\rbrace$ such that
$\sigma(i)$ and $\sigma(i+1)$ are coprime for $i=1,\ldots,n-1$.
All …
- 17.9k
3
votes
0
answers
90
views
Equirepartition of sums for large multisets in subsets of finite fields
Let
$p$ be a prime number and let $\mathcal A$ be a subset of $a\leq p$ distinct
elements in $\mathbb F_p$.
We denote by $\mathcal M_k(\mathcal A)$ the set of all ${k+a-1\choose k}$
multisets consisti …
- 17.9k
6
votes
0
answers
228
views
A bias for runs in Legendre symbols?
$\newcommand\Legendre[2]{\genfrac(){}{}{#1}{#2}}$An odd prime $p$ defines the sequence $\Legendre1 p,\Legendre2 p,\dotsc,\Legendre{p-1}p$
of values of the Legendre symbol describing the quadratic natu …
- 17.9k
2
votes
0
answers
92
views
Primes of the form $a+b^k$ for $k=(a \bmod 2),\ldots,n$?
Procrastinal problem: Given $n$, one can ask for integers $a,b>1$ of different parities
such that $a+b^k$ is prime for $k=(a\bmod 2),\ldots,n$.
A few examples are:
$2+4995825^k$ is prime for $k=0,\ld …
- 17.9k
1
vote
1
answer
276
views
There seem to be only few primes of the form ${n\choose k}+1$ if $k\geq 3$ is odd
We consider the sequence $n\longmapsto {n\choose k}+1$
for $k\geq 1$ a fixed integer. For $k\geq 3$ odd,
this sequence seems to contain surprisingly few prime numbers
while there are many primes (perh …
- 17.9k
14
votes
0
answers
294
views
An 'onion-structure' for roots of a series associated to prime numbers?
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the
sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers
defines a holomorphic function in the open disc of radius $e$.
…
- 17.9k
2
votes
2
answers
259
views
Inequalities for two functions related to the primorial function
Added: As remarked in the answers below, my question has a negative (and well-known) answer.
We denote by $\mathcal P=\lbrace 2,3,5,7,\ldots\rbrace$ the set of prime-numbers and by
$\mathcal P^*=\lbra …
- 17.9k
2
votes
0
answers
153
views
Electrostatic potential energy of point-charges at primes up to $x$
Given a positive real (or integral) number $x$ we consider the
electrostatic potential energy of equal point charges at all primes up to $x$
given by
$$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$
whe …
- 17.9k
1
vote
0
answers
164
views
Another Goldbach variation for odd numbers?
Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime.
Dabbling in the dark art …
- 17.9k
4
votes
1
answer
599
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of …
- 17.9k
10
votes
1
answer
312
views
Fixpoints of $m\longmapsto \mathrm{rad}(\phi(m^2))$ under iteration
Given a strictly positive integer $m$ let $\alpha(m)=\mathrm{rad}(m\phi(m))$
be the radical (product of all distinct prime divisors) of the product of $m$ and of Euler's totient function $\phi(m)=m\pr …
- 17.9k
6
votes
1
answer
388
views
Arithmetic properties of positively reduced $2\times 2$-matrices
Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$
positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces
at …
- 17.9k