en>TakuyaMurata: /* Vector flow in Lie group theory */

2014-12-06T08:05:46Z

Vector flow in Lie group theory

← Older revision		Revision as of 10:05, 6 December 2014
Line 1:		Line 1:
	~~In [[number theory]], '''Brocard's conjecture''' is a [[conjecture]] that there are at least four [[prime number]]s between (''p''<sub>''n''</sub>)<sup>2</sup>~~ and ~~(''p''<sub>''n''+1</sub>)<sup>2</sup>, for ''n'' > 1, where ''p''<sub>''n''</sub> is~~ the ~~''n''<sup>th</sup> prime number~~.~~<ref>{{mathworld\|urlname=BrocardsConjecture\|title=Brocard's Conjecture}}</ref> It~~ is ~~widely believed that this conjecture~~ is ~~true~~. ~~However, it remains unproven as of January 2014~~.		Greetings! I am Marvella and I really feel comfortable when individuals use the complete title. Hiring is my occupation. Doing ceramics is what love performing. California is our beginning place.<br><br>Also visit my weblog [http://www.streaming.iwarrior.net/blog/258268 over the counter std test]

	~~{\| class="wikitable" style="float:right; margin:10px;"~~
	\|-
	~~! n !!~~ <~~math~~>~~p_n~~<~~/math~~> ~~!! <math>p_n^2</math> !! Prime numbers !! <math>\Delta<~~/~~math>~~
	\|-
	~~\| 1 \|\| 2 \|\| 4 \|\| 5, 7 \|\| '''2'''~~
	\|-
	~~\| 2 \|\| 3 \|\| 9 \|\| 11, 13, 17, 19, 23 \|\| 5~~
	\|-
	~~\| 3 \|\| 5 \|\| 25 \|\| 29, 31, 37, 41, 43, 47 \|\| 6~~
	\|-
	~~\| 4 \|\| 7 \|\| 49 \|\| 53, 59, 61, 67, 71… \|\| 15~~
	\|-
	~~\| 5 \|\| 11 \|\| 121 \|\| 127, 131, 137, 139, 149… \|\| 9~~
	\|-
	~~\|colspan=5\| <math>\Delta</math> stands for <math>\pi(p_{n+1}^2) - \pi(p_n^2)<~~/~~math>.~~
	\|}

	~~The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27,~~ ... ~~{{OEIS2C\|id=A050216}}.~~

	[[Legendre's conjecture]] that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for ''p''<sub>''n''</sub> ≥ 3 since ''p''<sub>''n''+1</~~sub> - ''p''<sub>''n''<~~/~~sub> ≥ 2.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~{{Numtheory-stub}}~~

	~~[[Category:Conjectures about prime numbers]~~]

213.210.154.134: fixed link

2012-03-10T13:09:29Z

fixed link

New page

In [[number theory]], '''Brocard's conjecture''' is a [[conjecture]] that there are at least four [[prime number]]s between (''p''''n'')2 and (''p''''n''+1)2, for ''n'' > 1, where ''p''''n'' is the ''n''th prime number.<ref>{{mathworld|urlname=BrocardsConjecture|title=Brocard's Conjecture}}</ref> It is widely believed that this conjecture is true. However, it remains unproven as of January 2014.

{| class="wikitable" style="float:right; margin:10px;"
|-
! n !! <math>p_n</math> !! <math>p_n^2</math> !! Prime numbers !! <math>\Delta</math>
|-
| 1 || 2 || 4 || 5, 7 || '''2'''
|-
| 2 || 3 || 9 || 11, 13, 17, 19, 23 || 5
|-
| 3 || 5 || 25 || 29, 31, 37, 41, 43, 47 || 6
|-
| 4 || 7 || 49 || 53, 59, 61, 67, 71… || 15
|-
| 5 || 11 || 121 || 127, 131, 137, 139, 149… || 9
|-
|colspan=5| <math>\Delta</math> stands for <math>\pi(p_{n+1}^2) - \pi(p_n^2)</math>.
|}

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... {{OEIS2C|id=A050216}}.

[[Legendre's conjecture]] that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for ''p''''n'' ≥ 3 since ''p''''n''+1 - ''p''''n'' ≥ 2.

==Notes==
{{reflist}}

{{Numtheory-stub}}

[[Category:Conjectures about prime numbers]]

Vector flow - Revision history

en>TakuyaMurata: /* Vector flow in Lie group theory */

213.210.154.134: fixed link