Formula for primes - Revision history

en>David Eppstein: Undo — the part about Riemann hypothesis is important (otherwise we have no idea what A is, it's not just that we can't compute its higher digits)

2014-11-05T18:24:56Z

Undo — the part about Riemann hypothesis is important (otherwise we have no idea what A is, it's not just that we can't compute its higher digits)

← Older revision		Revision as of 20:24, 5 November 2014
Line 1:		Line 1:
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en>David Eppstein: Undo new section — unsourced, looks like original research

2014-02-23T17:42:13Z

Undo new section — unsourced, looks like original research

← Older revision		Revision as of 19:42, 23 February 2014
Line 1:		Line 1:
	~~[[Image:Pyramid of 35 spheres animation.gif\|frame\|right\|A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.]]~~		Dalton is what's written across his birth certificate but nevertheless , he never really enjoyed that name. The precious hobby for him and his kids is landscaping but he's been taking on new things a short time ago. Auditing is even his primary income originates from. Massachusetts is where he and his [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=wife+survive wife survive]. He's not godd at design but you might want to check their website: http://circuspartypanama.com<br><br>My web page clash of clans hack tool ([http://circuspartypanama.com navigate to this site])

	~~A '''tetrahedral number''~~', ~~or '''triangular pyramidal number''', is a [[figurate number]]~~ that ~~represents a [[pyramid (geometry)\|pyramid]] with a triangular base and three sides, called a [[tetrahedron]]~~. The ~~''n''th tetrahedral number~~ is ~~the sum of the first~~ '~~'n'' [[triangular number]]~~s.

	~~The first ten tetrahedral numbers are:~~

	:[[1 (number)\|1]], [[4 (number)\|4]], [[10 (number)\|10]], [[20 (number)\|20]], [[35 (number)\|35]], [[56 (number)\|56]], [[84 (number)\|84]], [[120 (number)\|120]], [[165 (number)\|165]], [[220 (number)\|220]], … {{OEIS\|id=A000292}}

	~~== Formula ==~~

	~~The formula for the ''n''-th tetrahedral number~~ is ~~represented by the 3rd [[rising factorial]] of ''n'' divided by the [[factorial]] of 3:~~
	~~:<math>T_n={n(n+1)(n+2)\over 6} = {n^{\overline 3}\over 3!}</math>~~

	~~The tetrahedral numbers can also be represented as [[binomial coefficient]]s:~~
	~~:<math>T_n={n+2\choose3}.</math>~~
	~~Tetrahedral numbers can therefore be found in the fourth position either~~ from ~~left or right in [[Pascal's triangle]]~~.

	~~== Geometric interpretation ==~~

	~~Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (''T''<sub>5</sub> = 35) can be modelled with 35 [[billiard ball]]s~~ and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

	~~When order-''n'' tetrahedra built from ''T''<sub>''n''</sub> spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest~~ [~~[sphere packing]] as long as ''n'' ≤ 4.<ref>http~~://~~www~~.~~pisquaredoversix~~.~~force9.co.uk~~/~~Tetrahedra.htm</ref>~~

	== ~~Properties~~ ==

	*[[A. ~~J. Meyl]] proved in 1878 that only three tetrahedral numbers are also [[Square number\|perfect squares]], namely:~~
	*:'~~'T''<sub>1</sub> = 1² = 1~~
	*:''T''<sub>2</sub> = 2² = 4
	*:''T''<sub>48</sub> = 140² = 19600.

	*[[Sir Frederick Pollock, 1st Baronet\|Sir Frederick Pollock]] conjectured that every number is the sum of at ~~most 5 tetrahedral numbers: see [[Pollock tetrahedral numbers conjecture]].~~

	* The only tetrahedral number that is also a [[square pyramidal number]] is 1 (Beukers, 1988), and the only tetrahedral number that is also a [[perfect cube]] is 1.

	* The [[infinite sum]] of tetrahedral numbers reciprocals is 3/2, which can be derived using [[telescoping series]]:
	*:<math> \!\ \sum_{n=1}^{\infty} \frac{6}{n(n+1)(n+2)} = \frac{3}{2}.</math>

	* The tetrahedron with basic length 4 (summing up to ~~20) can be looked at as the 3-dimensional analogue of the [[tetractys]], the 4th [[triangular number]] (summing up to 10).~~

	* The [[parity (mathematics)\|parity]] of tetrahedral numbers follows the repeating pattern odd-even-even-even.

	*An observation of tetrahedral numbers:
	*:~~''T''<sub>5<~~/~~sub> = ''T''<sub>4<~~/~~sub> + ''T''<sub>3</sub> + ''T''<sub>2</sub> + ''T''<sub>1</sub>~~

	*Numbers that are both triangular and tetrahedral must satisfy the [[binomial coefficient]] equation:
	*:<math>Tr_n={n+1\choose2}={m+2\choose3}=Te_m.<~~/math>~~

	* The only numbers that are both Tetrahedral and Triangular numbers are {{OEIS\|id=A027568}}:
	*: ''Te''<sub>1</sub> = ''Tr''<sub>1</sub> = 1
	*: ''Te''<sub>3</sub> = ''Tr''<sub>4</sub> = 10
	*: ''Te''<sub>8</sub> = ''Tr''<sub>15</sub> = 120
	*: ''Te''<sub>20</sub> = ''Tr''<sub>55<~~/sub~~> ~~= 1540~~
	*: ''Te''<sub>34</sub> = ''Tr''<sub>119</sub> = 7140

	~~==See also==~~
	* [[Centered triangular number]]

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	* {{MathWorld \|title=Tetrahedral Number \|urlname=TetrahedralNumber}}
	* [http://demonstrations.wolfram.com/GeometricProofOfTheTetrahedralNumberFormula/ Geometric Proof of ~~the Tetrahedral Number Formula] by Jim Delany, [[The Wolfram Demonstrations Project]].~~
	* [http://~~www.scribd~~.com~~/doc/50474167/On-the-Relation-Between-Summations-and-Tetrahedral-Numbers\| On the relation between double summations and tetrahedral numbers] by Marco Ripà~~

	~~{{Classes of natural numbers}}~~
	~~{{DEFAULTSORT:Tetrahedral Number}}~~
	~~[[Category:Figurate numbers]~~]

en>Duoduoduo: /* See also */ Generating primes

2013-10-18T16:28:36Z

See also: Generating primes

← Older revision		Revision as of 18:28, 18 October 2013
Line 1:		Line 1:
			[[Image:Pyramid of 35 spheres animation.gif\|frame\|right\|A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.]]

			A '''tetrahedral number''', or '''triangular pyramidal number''', is a [[figurate number]] that represents a [[pyramid (geometry)\|pyramid]] with a triangular base and three sides, called a [[tetrahedron]]. The ''n''th tetrahedral number is the sum of the first ''n'' [[triangular number]]s.

	~~Some people call me Gabrielle~~. ~~I am a cashier~~ and I'~~m preparing pretty good financially. As~~ a [http://www.~~Wonderhowto~~.~~com~~/~~search~~/~~definite+girl~~/ ~~definite girl~~] ~~what I simply like~~ is ~~going with karaoke but I in no way made~~ a ~~dime in addition to it~~. ~~My husband and I select~~ to ~~reside in Guam but I will have actually~~ to ~~move in a trustworthy year or two~~. ~~See what precisely new on my web blog here~~: ~~http~~://~~prometeu~~.~~net~~<br><br>~~Here is my blog how to hack clash~~ of ~~clans~~ [[http://~~prometeu~~.~~net simply click~~ the ~~next internet page~~]]		The first ten tetrahedral numbers are:

			:[[1 (number)\|1]], [[4 (number)\|4]], [[10 (number)\|10]], [[20 (number)\|20]], [[35 (number)\|35]], [[56 (number)\|56]], [[84 (number)\|84]], [[120 (number)\|120]], [[165 (number)\|165]], [[220 (number)\|220]], … {{OEIS\|id=A000292}}

			== Formula ==

			The formula for the ''n''-th tetrahedral number is represented by the 3rd [[rising factorial]] of ''n'' divided by the [[factorial]] of 3:
			:<math>T_n={n(n+1)(n+2)\over 6} = {n^{\overline 3}\over 3!}</math>

			The tetrahedral numbers can also be represented as [[binomial coefficient]]s:
			:<math>T_n={n+2\choose3}.</math>
			Tetrahedral numbers can therefore be found in the fourth position either from left or right in [[Pascal's triangle]].

			== Geometric interpretation ==

			Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (''T''<sub>5</sub> = 35) can be modelled with 35 [[billiard ball]]s and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

			When order-''n'' tetrahedra built from ''T''<sub>''n''</sub> spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest [[sphere packing]] as long as ''n'' ≤ 4.<ref>http://www.pisquaredoversix.force9.co.uk/Tetrahedra.htm</ref>

			== Properties ==

			*[[A. J. Meyl]] proved in 1878 that only three tetrahedral numbers are also [[Square number\|perfect squares]], namely:
			*:''T''<sub>1</sub> = 1² = 1
			*:''T''<sub>2</sub> = 2² = 4
			*:''T''<sub>48</sub> = 140² = 19600.

			*[[Sir Frederick Pollock, 1st Baronet\|Sir Frederick Pollock]] conjectured that every number is the sum of at most 5 tetrahedral numbers: see [[Pollock tetrahedral numbers conjecture]].

			* The only tetrahedral number that is also a [[square pyramidal number]] is 1 (Beukers, 1988), and the only tetrahedral number that is also a [[perfect cube]] is 1.

			* The [[infinite sum]] of tetrahedral numbers reciprocals is 3/2, which can be derived using [[telescoping series]]:
			*:<math> \!\ \sum_{n=1}^{\infty} \frac{6}{n(n+1)(n+2)} = \frac{3}{2}.</math>

			* The tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3-dimensional analogue of the [[tetractys]], the 4th [[triangular number]] (summing up to 10).

			* The [[parity (mathematics)\|parity]] of tetrahedral numbers follows the repeating pattern odd-even-even-even.

			*An observation of tetrahedral numbers:
			*:''T''<sub>5</sub> = ''T''<sub>4</sub> + ''T''<sub>3</sub> + ''T''<sub>2</sub> + ''T''<sub>1</sub>

			*Numbers that are both triangular and tetrahedral must satisfy the [[binomial coefficient]] equation:
			*:<math>Tr_n={n+1\choose2}={m+2\choose3}=Te_m.</math>

			* The only numbers that are both Tetrahedral and Triangular numbers are {{OEIS\|id=A027568}}:
			*: ''Te''<sub>1</sub> = ''Tr''<sub>1</sub> = 1
			*: ''Te''<sub>3</sub> = ''Tr''<sub>4</sub> = 10
			*: ''Te''<sub>8</sub> = ''Tr''<sub>15</sub> = 120
			*: ''Te''<sub>20</sub> = ''Tr''<sub>55</sub> = 1540
			*: ''Te''<sub>34</sub> = ''Tr''<sub>119</sub> = 7140

			==See also==
			* [[Centered triangular number]]

			==References==
			{{Reflist}}

			==External links==
			* {{MathWorld \|title=Tetrahedral Number \|urlname=TetrahedralNumber}}
			* [http://demonstrations.wolfram.com/GeometricProofOfTheTetrahedralNumberFormula/ Geometric Proof of the Tetrahedral Number Formula] by Jim Delany, [[The Wolfram Demonstrations Project]].
			* [http://www.scribd.com/doc/50474167/On-the-Relation-Between-Summations-and-Tetrahedral-Numbers\| On the relation between double summations and tetrahedral numbers] by Marco Ripà

			{{Classes of natural numbers}}
			{{DEFAULTSORT:Tetrahedral Number}}
			[[Category:Figurate numbers]]

en>Bibcode Bot: Adding 3 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-08-29T17:03:33Z

Adding 3 arxiv eprint(s), 0 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

New page

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