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.
2,216 questions
6
votes
0
answers
144
views
Status of Mills' constant
There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
-1
votes
0
answers
59
views
Are 6n ± 1 or more general arithmetic progression forms used as a core tool in advanced Number Theory proofs, beyond basic sieving? [closed]
Is there any research that uses the 6n ± 1 form or the more general form kn ± r to prove more important prime number theorems? (e.g., linked to Dirichlet's theorem on arithmetic progressions)
3
votes
1
answer
226
views
On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18k
1
vote
1
answer
148
views
On even numbers of the form $p+p'+2^k$
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18k
-1
votes
1
answer
166
views
Whether $2n>10$ can be written as $p+p'+2^a+2^b$ with $p$ and $p'$ consecutive primes?
In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...
- 18k
3
votes
1
answer
125
views
Sum of prime divisors functions
I was idly thinking today about the functions $\displaystyle f(n) = \sum_{p \mid n} p$ and $\displaystyle F(n) = \sum_{p^e \| n} ep$, respectively the "sum of prime divisors" function and ...
- 101
4
votes
0
answers
117
views
Character and exponential sums over primes under GRH
Theorem 18.13 of Montgomery and Vaughan's multiplicative number theory book says (ignoring $\log $'s) $$\int _{-\delta }^{\delta }|\psi _\chi (\beta )|^2d\beta \ll \delta x\hspace {15mm}\psi _\chi (\...
- 1,656
4
votes
0
answers
288
views
Congruence restrictions on prime divisors of polynomial values
Consider the polynomial $f(x)= x^2+1$. Can you prove that there are infinitely many integers $x$ such that $f(x)$ has no prime divisor congruent to $1 \bmod 3$? Obviously the prime divisors are ...
8
votes
1
answer
622
views
Do all primes $>2$ hit $5$?
$2$ is a fixed point of the iteration:
$$q_{n+1}:=\min_{p|(q_n-1)^2+1} p$$
Start with $q_1>2$ prime. Does this iteration hit $5$? (min runs over primes)
- 3,761
5
votes
0
answers
335
views
For a prime, is there always a prime number for which it is a primitive root?
Artin’s primitive root conjecture states that for any integer $a\neq \pm1$ which is not a square,there are infinitely many primes $p$ such that $a$ is a primitive root mod $p$. By Heath-Brown's result,...
- 528
2
votes
0
answers
91
views
Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
- 18k
4
votes
1
answer
438
views
Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?
The groundbreaking work of Maynard and Tao showed the following fundamental result:
For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at ...
- 43
7
votes
0
answers
297
views
Is 1001 the only palindrome which is a product of three consecutive primes?
I made a computational search for over all integers $N < 10^{27}$.
Method:
Generate a list of primes up to $10^9$
Iterate over consecutive prime triples and compute the product
Check each product ...
- 115
21
votes
2
answers
2k
views
Can every natural number be written as a product of integer powers of primes minus 1?
Do for every natural number $n$ exist (possibly negative) integers $a_p$, finitely many of them nonzero, such that
$$\log(n) = \sum_{p \text{ prime}} a_p \log(p-1)\,?$$
Equivalently:
$$n = \prod_{p \...
- 3,761
-5
votes
1
answer
192
views
Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ — minimal of this form — does the number $k = (q-1)/p$ have only prime divisors $<p$?
Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ - minimal of this form -,
does then the number $k = (q-1)/p$ have only prime divisors $< p$?
What does the research literature say for ...
- 3,761