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 not deleted
user 37555
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.
22
votes
Accepted
What keeps asymptotic Goldbach's conjecture out of reach of current technology?
As far as I know there are two approaches to Goldbach type problems, the circle method and sieve methods. In the sequel I will restrict myself to the circle method, hoping that someone else writes som …
17
votes
Accepted
Number of primes with $-1\pmod 6$ vs. Number of primes with $+1\pmod 6$
Assume that the Riemann hypothesis for the non-principal $L$-series $\pmod{3}$ is false, say, this series has a zero $\rho=\sigma+i\gamma$ with $\sigma>1/2$. Then Turan and Knapowski have shown that b …
15
votes
Accepted
Does this prime-gaps pattern occur infinitely often?
This would follow from the $k$-tuple conjecture in the following way.
Choose an admissible tuple $d_1, \ldots, d_{k+2}$, such that $d_2-d_1=d_{k+2}-d_2$. If $n\in[d_1, d_{k+2}]$ is an integer, such t …
14
votes
Understanding Vaughan's Identity
The analytic version of Vaughan's identity is
$$
\frac{\zeta'}{\zeta} = F+\zeta'G-FG\zeta + \left(\frac{\zeta'}{\zeta}-F\right)(1-\zeta G).
$$
Here the last factor to the right is the most complicated …
12
votes
0
answers
626
views
Sieve bound for prime $k$-tuples
Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, d_k) (Ck) …
11
votes
1
answer
408
views
Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[x, …
10
votes
Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?
The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …
10
votes
Is Li(x) the best possible approximation to the prime-counting function?
Whether for a finite set $\mathcal{R}$ of roots the approximation
$$
\pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho)
$$
is "on average" better …
10
votes
Can anything deep be said uniformly about conjectures like Goldbach's?
There exist surprising counterexamples. Elsholtz and Dietmann found the following: If $p\equiv 7\pmod{8}$ is prime, then the equation $x^2+y^2+z^4=p^2$ has no non-trivial solution. You might argue tha …
8
votes
Primality test for $2p+1$
The if-part is a special case of Pocklingtons theorem (see https://en.wikipedia.org/wiki/Pocklington_primality_test ). The only if part requires some computation, as you need a quadratic non-reside mo …
7
votes
Accepted
Two equivalent statements about primes
Statement a) is true for all $N$. This follows from the fact that the number of even integers which cannot be expressed as the sum of 2 primes is small. The best result in this direction is due to Pin …
6
votes
Accepted
Prime factors of the members of a certain recurrence
It is even true that the equation $n^2+1=m$ with $p|m\Rightarrow p\in S$ has only finitely many solutions. To see this note that every $m$ satisfying this property can be written as $m=m_1m_2^3$, wher …
6
votes
Quadratic residues and nonresidues of arbitrary patterns
The number of integers $a$ in $[0, x]$ with the desired property is
$$
2^{-n}\sum_{a=1}^x\prod_{i=1}^n\left(1+\epsilon_i\left(\frac{a}{p_i}\right)\right).
$$
Expand the right hand side to obtain one t …
5
votes
Accepted
Specializing non-trivial primality tests
Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that …
5
votes
Accepted
Unknown bias in a distribution related to prime numbers
The number of totient divisors of $n$ is $d(n-1)-d((n-1, \varphi(n))$. As $n$ gets large, then almost all $n$ have the property that $\varphi(n)$ is divisble by all small primes. The average number of …