Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with prime-numbers
Search options answers only
not deleted
user 12176
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
3-7 primes in base 10
Significant progress on this question has been made by Maynard. In this preprint on the arXiv he proves that:
"There are infinitely many primes that do not contain the digit $9$ in their decimal …
- 11.4k
4
votes
Better error bounds for partial sums of reciprocals of primes?
Up to a factor of logarithm, $E(x)$'s oscillation has an amplitude which is of the same magnitude as that of $\frac{1}{x}\left(\psi(x)-x\right)$, that is the error in the prime number theorem. Specifi …
- 11.4k
13
votes
Accepted
How big can a set of integers be if all pairs have small gcd?
We'll prove that the maximal cardinality of such a set for $M^2\leq N$ has size equal to $$\pi(N)+\sum_{1<n\leq M} \pi(p(n))$$ where $p(n)$ is the smallest prime factor of $n$. Since $$\sum_{1<n\le …
- 11.4k
6
votes
Does the Maynard-Tao Theorem apply to general tuples of linear forms?
Note: This question was answered in the comments, and I am posting to remove the question from the unanswered list.
The result of Maynard and Tao applies to all admissible $k$-tuples of linear forms. …
- 11.4k
22
votes
Accepted
On prime numbers
The sums above are Riemann sums over different partitions for the function $\arctan(x)$ between $x=0$ and $x=1$, so the limit will equal the integral which is $\frac{\pi}{4}$. Both of these converge …
- 11.4k
4
votes
Density of a set of integers
I just wanted to add, we also have a similar upper bound which combined with unknown(google)'s answer shows that
$$E_{r}(x)\asymp\frac{x\left(\log\log x\right)^{r}}{\left(\log x\right)^{\frac{1}{2}} …
- 11.4k
4
votes
Accepted
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
Your above identity stems from $$\text{li(x)}-\Pi(x)+\text{small}=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty} \log \left((s-1)\zeta(s)\right)\frac{x^s}{s}ds,$$ and a Taylor expansion of the logarithm …
- 11.4k
41
votes
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
Edit (20/11/2013) : Yesterday James Maynard posted the paper Small gaps between primes on the arxiv in which he shows that for any $m$ there exists a constant $C_m$ such that $$ p_{n+m}-p_n\leq C_m$$ …
- 11.4k
40
votes
Accepted
What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that ext...
The major difference is the choice of the Selberg Sieve weights. I strongly recommend reading Maynard's paper. It is well written, and the core ideas are nicely explained.
In what follows, I'll gi …
- 11.4k
12
votes
Accepted
The tightest prime zipper
The slowest growing zipper will depend on the size of $p_{n+1}-p_n$ where $p_n$ is the $n^{th}$ prime number. There are many results regarding the size of the largest prime gap.
Unconditional: The w …
- 11.4k
14
votes
Median largest-prime-factor
I posted a paper to the arXiv which deals with this question along with some other things.
Using some results regarding either the mean of $\omega(n)$, or integers without large prime factors, we can …
- 11.4k
19
votes
Accepted
Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\...
Yes, and in fact there will still be many primes even if the size of the sector decrease to zero quite quickly.
In the 2001 paper "Gaussian primes in narrow sectors," Harman and Lewis use sieve meth …
- 11.4k
6
votes
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I am not exactly sure what you mean by bounds for $L/R$, but I assume that this translates to the rate of convergence and the next terms in the asymptotic. In this short note, a more precise asymptoti …
- 11.4k
7
votes
Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
For $\alpha$ fixed, this is relatively easy to prove. I leave it to you deduce the desired result as a consequence of the following theorem:
Theorem: Let $x>0$ and fix $0<\alpha<1$. Define $\omega …
- 11.4k
25
votes
Accepted
Why is the Chebyshev function relevant to the Prime Number Theorem
There are several ideas here, some mentioned in the other answers:
One: When Gauss was a boy (by the dates found on his notes he was approximately 16) he noticed that the primes appear with density …
- 11.4k