Number - User contributions [en]

MediaWiki:Sitenotice

2024-10-06T23:15:14Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-10-06T23:14:36Z

Vipul:

* <math>\sqrt{7 + 2}!! + 4 = 724</math>
* <math>\sqrt{7 + \sqrt{4}}!! + 4! = 744</math>
* <math>5^{1 + 2} = 125</math>
* <math>6^{2 + 1} = 216</math>
* <math>(3 + 4)^3 = 343</math>

MediaWiki:Sitenotice

2024-09-30T01:27:06Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-09-06T01:31:25Z

Vipul:

Number:Enabling site search autocompletion

2024-09-06T01:30:49Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Number). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in th..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Number).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

File:Site search autocompletion working.png

2024-09-06T01:30:37Z

Vipul:

File:Site search autocompletion broken.png

2024-09-06T01:30:18Z

Vipul:

Number:429 Too Many Requests error

2024-09-06T01:27:05Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

Vipul: /* Measures of difference */

Euler totient function

2014-01-29T22:03:42Z

Vipul:

2014-01-29T21:30:00Z

Vipul: /* Related facts and conjectures */

==Statement==

Suppose <math>f_1,f_2,\dots,f_r</math> are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product <math>\prod_{i=1}^r f_i</math> does not have any ''fixed divisors'', i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. '''Schinzel's hypothesis H''' states that there are infinitely many natural numbers <math>n</math> satisfying the condition that <math>f_1(n), f_2(n), \dots, f_r(n)</math> are all ''simultaneously'' prime.

==Related facts and conjectures==

===Weaker facts and conjectures===

{| class="sortable" border="1"
! Fact or conjecture !! Status !! How it fits with Schinzel's hypothesis H
|-
| [[Dickson's conjecture]] || Open || We are dealing with linear polynomials.
|-
| [[Bunyakovsky conjecture]] || open ||We are dealing with only one irreducible polynomial of degree two or higher
|-
| [[Dirichlet's theorem on primes in arithmetic progressions]] || proved || We are dealing with one irreducible polynomial of degree one
|-
| [[Twin prime conjecture]] || open || We are dealing with the irreducible polynomials <math>x</math> and <math>x + 2</math>. Also, the dependence is via [[Dickson's conjecture]].
|-
| [[Green-Tao theorem]] || proved || The theorem states that the sequence of primes contains arithmetic progressions of arbitrary length. The Green-Tao theorem can be viewed as a corollary of Schinzel's hypothesis H if we view it as a collection of statements, one about the existence of arithmetic progressions of a specific length. Each such statement is ''substantially'' weaker than what we'd get from Schinzel's hypothesis H (which not only guarantees the existence of such arithmetic progressions, but also allows us to choose a common difference for the arithmetic progression that satisfies some divisibility conditions). The dependence is via [[Dickson's conjecture]].
|}

Dickson's conjecture

2014-01-29T21:29:45Z

Vipul: Created page with "==Statement== Suppose <math>a_1,a_2,\dots,a_k,b_1,b_2,\dots,b_k</math> are integers with all the <math>a_i \ge 1</math>. Then, consider the polynomials: <math>f_i(x) := a_ix..."

Schinzel's hypothesis H

2014-01-29T21:26:35Z

Vipul: /* Weaker facts */

==Statement==

Suppose <math>f_1,f_2,\dots,f_r</math> are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product <math>\prod_{i=1}^r f_i</math> does not have any ''fixed divisors'', i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. '''Schinzel's hypothesis H''' states that there are infinitely many natural numbers <math>n</math> satisfying the condition that <math>f_1(n), f_2(n), \dots, f_r(n)</math> are all ''simultaneously'' prime.

==Related facts and conjectures==

===Weaker facts===

{| class="sortable" border="1"
! Fact or conjecture !! Status !! How it fits with Schinzel's hypothesis H
|-
| [[Dickson's conjecture]] || Open || We are dealing with linear polynomials.
|-
| [[Bunyakovsky conjecture]] || open ||We are dealing with only one irreducible polynomial of degree two or higher
|-
| [[Dirichlet's theorem on primes in arithmetic progressions]] || proved || We are dealing with one irreducible polynomial of degree one
|-
| [[Twin prime conjecture]] || open || We are dealing with the irreducible polynomials <math>x</math> and <math>x + 2</math>. Also, the dependence is via [[Dickson's conjecture]].
|-
| [[Green-Tao theorem]] || proved || The theorem states that the sequence of primes contains arithmetic progressions of arbitrary length. The Green-Tao theorem can be viewed as a corollary of Schinzel's hypothesis H if we view it as a collection of statements, one about the existence of arithmetic progressions of a specific length. Each such statement is ''substantially'' weaker than what we'd get from Schinzel's hypothesis H (which not only guarantees the existence of such arithmetic progressions, but also allows us to choose a common difference for the arithmetic progression that satisfies some divisibility conditions). The dependence is via [[Dickson's conjecture]].
|}

Schinzel's hypothesis H

2014-01-29T21:23:57Z

Vipul: /* Weaker facts */

==Statement==

Suppose <math>f_1,f_2,\dots,f_r</math> are all irreducible polynomials with integer coefficients and with positive leading coefficient, such that the product <math>\prod_{i=1}^r f_i</math> does not have any ''fixed divisors'', i.e., it cannot be expressed as a proper multiple of an integer-valued polynomial. '''Schinzel's hypothesis H''' states that there are infinitely many natural numbers <math>n</math> satisfying the condition that <math>f_1(n), f_2(n), \dots, f_r(n)</math> are all ''simultaneously'' prime.

==Related facts and conjectures==

===Weaker facts===

{| class="sortable" border="1"
! Fact or conjecture !! Status !! How it fits with Schinzel's hypothesis H
|-
| [[Bunyakovsky conjecture]] || open ||We are dealing with only one irreducible polynomial of degree two or higher
|-
| [[Dirichlet's theorem on primes in arithmetic progressions]] || proved || We are dealing with one irreducible polynomial of degree one
|-
| [[Twin prime conjecture]] || open || We are dealing with the irreducible polynomials <math>x</math> and <math>x + 2</math>
|-
| [[Green-Tao theorem]] || proved || The theorem states that the sequence of primes contains arithmetic progressions of arbitrary length. The Green-Tao theorem can be viewed as a corollary of Schinzel's hypothesis H if we view it as a collection of statements, one about the existence of arithmetic progressions of a specific length. Each such statement is ''substantially'' weaker than what we'd get from Schinzel's hypothesis H (which not only guarantees the existence of such arithmetic progressions, but also allows us to choose a common difference for the arithmetic progression that satisfies some divisibility conditions).
|}

Schinzel's hypothesis H

2014-01-29T21:19:28Z

Vipul: /* Related facts and conjectures */

Prime gap

2014-01-29T21:12:01Z

Vipul: /* Conjectures and advanced facts on upper bounds on limit inferior and occurrence of small prime gaps */

==Definition==

The '''prime gap''' between a prime <math>p</math> and its successor prime <math>q</math> is the difference <math>q - p</math>. In other words, a prime gap is a gap between two successive primes.

==Facts==

We are interested in three broad things:

* How frequently does a given prime gap occur?
* The limit inferior of prime gaps, i.e., as we go to infinity, what is the limiting value of smallest prime gaps?
* The limit superior of prime gaps, i.e., as we go to infinity, what is the limiting value of largest prime gaps?

===Basic facts (lower bound on limit inferior, limit superior is infinity)===

* A prime gap of <math>1</math> occurs between <math>2</math> and <math>3</math>, and never again. All other prime gaps are even, and at least <math>2</math>.
* [[There exist arbitrarily large prime gaps]]: This is because there exist arbitrarily large sequences of consecutive composite integer. For instance, for any <math>n > 1</math>, the sequence <math>n! + 2, n! + 3, \dots, n! + n</math> is a sequence of composite integers.

===Conjectures and advanced facts on upper bounds on limit inferior and occurrence of small prime gaps===

{| class="sortable" border="1"
! Name of conjecture/fact !! Statement !! Status
|-
| [[twin prime conjecture]] || There exist arbitrarily large pairs of [[twin primes]] -- successive primes with a gap of two. Equivalently, the limit inferior of prime gaps is exactly 2. || open
|-
| [[Polignac's conjecture]] || For any natural number <math>n</math>, the prime gap <math>2n</math> occurs for arbitrarily large pairs of primes || open; stronger than twin primes conjecture
|-
| [[Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]] || there exist infinitely many pairs of consecutive primes with prime gap at most 16 || proof conditional to [[Elliott-Halberstam conjecture]]
|-
| [[Zhang's theorem on bounded prime gaps]] || There exist infinitely many pairs of primes that differ by at most 70 million. In other words, there eixsts a prime gap <math>m</math> that is at most 70 million that occurs infinitely often. Subsequent work by the Polymath project brought the bound down to 4680|| Closed, see [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ here] and [http://www.michaelnielsen.org/polymath1/index.php?title=Timeline_of_prime_gap_bounds here].
|-
| [[Maynard's theorem on bounded prime gaps]] || There exist infinitely many pairs of primes that differ by at most 600. Subsequent work brought the bound done to 270. || Closed, see [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ here] and [http://www.michaelnielsen.org/polymath1/index.php?title=Timeline_of_prime_gap_bounds here].
|}

===Conjectures and advanced facts on maximum prime gaps (growth of limit superior relative to size of numbers)===

{| class="sortable" border="1"
! Name of conjecture/fact !! Statement !! Function (big-O) !! Status
|-
| [[Cramér's prime gap conjecture]] || For any prime <math>p</math>, the prime gap between <math>p</math> and the next prime is at most <math>c(\log p)^2</math>, <math>c</math> fixed || <math>O((\log p)^2)</math> || open
|-
| [[Prime-between-squares conjecture]] || There exists a prime between any two successive squares. Puts upper bound of <math>1 + 2\sqrt{p}</math> on prime gap || <math>O(\sqrt{p})</math> ||open
|-
| (corollary of) [[Generalized Riemann hypothesis]] || The prime gap between a prime <math>p</math> and the next prime is <math>O(\sqrt{p} (\log p))</math> || <math>O(\sqrt{p} \log p)</math> || open
|-
| [[exponent bound for prime gap of 0.535]] || The prime gap between <math>p</math> and the next prime is at most <math>p^{0.535}</math> || <math>O(p^{0.535})</math> || proved
|-
| (corollary of) [[prime number theorem]] || there exists a prime between <math>n</math> and <math>\alpha n</math> for any <math>\alpha > 1</math>, for <math>n</math> large enough (dependent on <math>\alpha</math>) || <math>O(n)</math> || proved
|-
| [[Bertrand's postulate]] || there exists a prime between <math>n</math> and <math>2n</math> ||<math>O(n)</math> || proved
|}

===Other related facts===

* [[Chen's theorem on primes and semiprimes with fixed separation]]