Prime gap - Revision history

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

2014-01-29T21:12:01Z

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

← Older revision		Revision as of 21:12, 29 January 2014
Line 21:		Line 21:
	! Name of conjecture/fact !! Statement !! Status		! Name of conjecture/fact !! Statement !! Status
	\|-		\|-
	\| [[twin ~~primes~~ 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		\| [[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		\| [[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
Line 27:		Line 27:
	\| [[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]]		\| [[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]		\| [[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. \|\| Closed, see [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ 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].
	\|}		\|}

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

2014-01-28T17:51:01Z

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

← Older revision		Revision as of 17:51, 28 January 2014
Line 21:		Line 21:
	! Name of conjecture/fact !! Statement !! Status		! Name of conjecture/fact !! Statement !! Status
	\|-		\|-
	\| [[twin primes conjecture]] \|\| ~~there~~ exist arbitrarily large pairs of [[twin primes]] -- successive primes with a gap of two. \|\| open		\| [[twin primes 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		\| [[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]]		\| [[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]
			\|-
			\| [[Maynard's theorem on bounded prime gaps]] \|\| There exist infinitely many pairs of primes that differ by at most 600. \|\| Closed, see [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ here]
	\|}		\|}

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

2010-05-02T01:53:18Z

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

← Older revision		Revision as of 01:53, 2 May 2010
Line 25:		Line 25:
	\| [[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		\| [[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 ~~condition~~ to [[Elliott-Halberstam 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]]
	\|}		\|}

Vipul at 01:52, 2 May 2010

2010-05-02T01:52:49Z

← Older revision		Revision as of 01:52, 2 May 2010
Line 5:		Line 5:
	==Facts==		==Facts==

	We are interested in ~~two~~ broad things:		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 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?		* The limit superior of prime gaps, i.e., as we go to infinity, what is the limiting value of largest prime gaps?
Line 15:		Line 16:
	* [[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.		* [[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===		===Conjectures and advanced facts on upper bounds on limit inferior and occurrence of small prime gaps===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
Line 21:		Line 22:
	\|-		\|-
	\| [[twin primes conjecture]] \|\| there exist arbitrarily large pairs of [[twin primes]] -- successive primes with a gap of two. \|\| open		\| [[twin primes conjecture]] \|\| there exist arbitrarily large pairs of [[twin primes]] -- successive primes with a gap of two. \|\| 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 condition to [[Elliott-Halberstam conjecture]]
	\|}		\|}

Vipul at 02:50, 9 February 2010

2010-02-09T02:50:52Z

← Older revision		Revision as of 02:50, 9 February 2010
Line 5:		Line 5:
	==Facts==		==Facts==

	===Basic facts===		We are interested in two broad things:

			* 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>.		* 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.		* [[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 ~~minimum prime gaps~~===		===Conjectures and advanced facts on upper bounds on limit inferior===

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
Line 18:		Line 23:
	\|}		\|}

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

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul at 02:46, 9 February 2010

2010-02-09T02:46:47Z

← Older revision		Revision as of 02:46, 9 February 2010
Line 35:		Line 35:
	\| [[Bertrand's postulate]] \|\| there exists a prime between <math>n</math> and <math>2n</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]]

Vipul: /* Conjectures and advanced facts on maximum prime gaps */

2010-02-09T01:04:13Z

Conjectures and advanced facts on maximum prime gaps

← Older revision		Revision as of 01:04, 9 February 2010
Line 29:		Line 29:
	\| (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		\| (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 <math>O(p^{0.535})</math> \|\| <math>O(p^{0.535})</math> \|\| proved		\| [[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		\| (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

Vipul: /* Conjectures and advanced facts on maximum prime gaps */

2010-02-09T00:59:48Z

Conjectures and advanced facts on maximum prime gaps

← Older revision		Revision as of 00:59, 9 February 2010
Line 23:		Line 23:
	! Name of conjecture/fact !! Statement !! Function (big-O) !! Status		! 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		\| [[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		\| [[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
Line 33:		Line 33:
	\| (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		\| (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		\| [[Bertrand's postulate]] \|\| there exists a prime between <math>n</math> and <math>2n</math> \|\|<math>O(n)</math> \|\| proved
	\|}		\|}

Vipul: Created page with '==Definition== The '''prime gap''' between a prime $p$ and its successor prime $q$ is the difference $q - p$ . In other words, a prime gap is a g…'

2010-02-09T00:59:21Z

Created page with '==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 g…'

New page

==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==

===Basic facts===

* 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 minimum prime gaps===

{| class="sortable" border="1"
! Name of conjecture/fact !! Statement !! Status
|-
| [[twin primes conjecture]] || there exist arbitrarily large pairs of [[twin primes]] -- successive primes with a gap of two. || open
|}

===Conjectures and advanced facts on maximum prime gaps===

{| 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 <math>O(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
|}

Prime gap - Revision history

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

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

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

Vipul at 01:52, 2 May 2010

Vipul at 02:50, 9 February 2010

Vipul at 02:46, 9 February 2010

Vipul: /* Conjectures and advanced facts on maximum prime gaps */

Vipul: /* Conjectures and advanced facts on maximum prime gaps */

Vipul: Created page with '==Definition== The '''prime gap''' between a prime p and its successor prime q is the difference q - p. In other words, a prime gap is a g…'

Vipul: Created page with '==Definition== The '''prime gap''' between a prime $p$ and its successor prime $q$ is the difference $q - p$ . In other words, a prime gap is a g…'