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Questions tagged [prime-numbers]

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.

2020 questions
31
votes
4 answers

What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this. Permit me to define prime number the following way: A number $n>1$ is a prime if all integers $d$ with $1< d \leq \sqrt{n}$ give non-zero remainders …
P Vanchinathan
  • 2,543
11
votes
1 answer

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 asymptotically an arithmetic progression.…
Roland Bacher
  • 17,432
9
votes
2 answers

Cubic polynomial mapping primes to primes

Let $f(n)=a_3n^3+a_2n^2+a_1n+a_0$, with $a_i\in\mathbb{Z}$, $a_3>0, a_0\neq 0$ such that $f(n)>0$ for all positive integers $n$. Given a prime $p$, when is $f(p)$ again prime? For example, let $f(n)=7n^3-50n+30$. Then, $$f(7)=2081\quad {\rm…
Jeff Breeding-Allison
  • 315
9
votes
3 answers

question in prime numbers

Is it true that in any successive (natural) $2p_n$ numbers there are at least three numbers that are not divisible by any prime less (not equal) than $p_n$? Here, $p_n$ denotes the $n$-th prime number. For example in any six successive numbers…
Asterios Gkantzounis
  • 581
  • 5
  • 27
8
votes
3 answers

special primes with p'=4p+1

How can I most quickly find a big prime, p, for which 4p+1 is also prime? For example, p=37 works. I wonder if these special primes have been researched and some characteristics are known. Are there infinitely many of these primes?
bobuhito
  • 1,537
7
votes
0 answers

Does the primality of the number of dimensions affect its properties?

Motivated by How does the parity of a dimension affect its properties? I dare to ask the following question (with thanks to my colleague Vedran Dunjko): We happen to live in a world of prime dimensionality. Is that special? Are there properties of…
Carlo Beenakker
  • 177,695
5
votes
2 answers

Extended Euclid proof and primes in form $|\prod\limits_{n \neq m} p_n -\prod\limits_{m \neq n} p_m|$

Euclid proof of the infinitude of primes can be extended into this. Assuming there is a finite number of primes, $k$, sort them in increasing order and split the series after any prime at $t$. Create the difference between the products of each…
user113386
5
votes
1 answer

The position of a prime between the two neighboring primes

Let $$ p_0:=2\, \ldots\, p_{n-1}\,\,p_n\,\,p_{n+1}\,\ldots $$ be the increasing sequence of all primes. How often (three questions): $\,p_n > \frac{p_{n-1}+p_{n+1}}2 $ $\,\sqrt{p_{n-1}\cdot p_{n+1}}\, <\, p_n\, \le\, \frac{p_{n-1}+p_{n+1}}2 $ $\,…
Wlod AA
  • 4,686
  • 16
  • 23
5
votes
0 answers

Gaps Between Primes in Arithmetic Progression

I'm interested in knowing if there exists a bound (possibly similar to the one found by Baker, Harman, and Pintz) for gaps between primes in arithmetic progression given a fixed quadratic character. For example, does there exist an $\epsilon > 0$…
The Traveling Salesman
  • 109
5
votes
1 answer

The Nth number with M prime factors

Hi. Suppose we arrange all natural numbers in a matrix P defined as follows: P[I][J] = The Jth number with I prime factors. So P looks something like: 1 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , …
barak manos
  • 595
4
votes
1 answer

form of primes:prime plus a power of 2?

is every prime p equals another prime p' plus or minus a power of 2? p=p'+/-2^n? are there infinitely many primes not of this form?
Asterios Gkantzounis
  • 581
  • 5
  • 27
3
votes
3 answers

prime powers between n and 2n

Is there a result in the spirit of Bertrand-Chebyshev which talks about the existence of prime powers between n and 2n (or 3n or something like that) for n large?
Chebolu
  • 575
3
votes
1 answer

Work down on the Upper bound of the Twin Primes

It can be shown using the Selberg Sieve method, that the maximum number of Twin primes less than $N$ is $$\frac{CN}{\ln^2(N)}$$ does anyone know if there has been any work done on finding an upper bound for the constant $C$?
Alex Botros
  • 269
3
votes
2 answers

Does there exist an even positive integer $n$ such that, for each prime number $p>2$, $p+n$ is not a prime number?

I don't know if this is a known problem, but I didn't find any similar question. Let's do some example to explain what I'm searching. Take $n=10$. We have $p=3$ odd prime number and also $p+n = 3+10=13$ prime, so $n=10$ is not valid Take $n=30$. We…
Laransoft
  • 31
3
votes
0 answers

Generalizing Mersenne prime search to cyclotomic prime

The Mersenne prime $M_p=2^p-1$ can be expressed as $M_p=\Phi_p(2)$ where $\Phi_m(x)$ is a cyclotomic polynomial. It seems useful to generalize the Mersenne prime search to more general cyclotomic primes. Let $\Phi_m(x,y)=\Phi_m(x/y)y^{\phi(m)}$ be…
CHUAKS
  • 964
1
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