Formula for primes: Difference between revisions

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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)
 
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[[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.]]
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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>&nbsp;=&nbsp;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''&nbsp;≤&nbsp;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]]

Latest revision as of 19:24, 5 November 2014

You might discover simple methods to accelerate computer by creating the most from the built in tools in your Windows also as downloading the Service Pack updates-speed up a PC plus fix error. Simply follow a limited guidelines to quickly create a computer swiftly than ever.

Windows Defender - this does come standard with several Windows OS Machines, however, otherwise may be download from Microsoft for free. It usually help protect against spyware.

Perfect Optimizer also offers to remove junk files and is completely Windows Vista compatible. Most registry product just don't have the time plus funds to analysis Windows Vista mistakes. Because ideal optimizer has a large customer base, they do have the time, cash and reasons to support completely help Windows Vista.

Your computer was rather swiftly whenever you initially bought it. Because a registry was extremely clean and free of errors. After time, the computer begins to run slow and freezes up today plus then. Because there are mistakes accumulating in it and certain information is rewritten or even completely deleted by your wrong uninstall of programs, wrong operations, malware or additional elements. That is the reason why your computer performance decreases slowly plus become fairly unstable.

Another thing we should check is whether the tuneup utilities 2014 system which you are considering has the ability to detect files plus programs which are superior. One of the registry cleaner programs you may try is RegCure. It is helpful for speeding up and cleaning up difficulties on a computer.

Your system is made and built for the purpose of helping you accomplish tasks plus not be pestered by windows XP error messages. When there are mistakes, what do we do? Some people pull their hair and cry, whilst those sane ones have their PC repaired, while those actually wise ones analysis to have the errors fixed themselves. No, these errors were not additionally tailored to rob you off your cash and time. There are details to do to really prevent this from happening.

Why this really is significant, is considering most 'dumb' registry products really delete these files without even recognizing. They simply browse from the registry and try plus discover the many problems possible. They then delete any files they see fit, and considering they are 'dumb', they don't actually care. This means which when they delete several of these vital system files, they are actually going to result a LOT more harm than superior.

So, the number one thing to do whenever a computer runs slow is to buy an authentic plus legal registry repair tool that would help you eliminate all difficulties associated to registry and enable you enjoy a smooth running computer.

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