Preference learning - Revision history

en>Monkbot: /* References */Task 2: Fix CS1 deprecated coauthor parameter errors

2014-07-22T17:55:25Z

References: Task 2: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 19:55, 22 July 2014
Line 1:		Line 1:
	~~[[File:StellaOctangulaNumber.jpg\|thumb\|240px\|124 [[Neodymium magnet toys\|magnetic balls]] arranged into the shape of a [[stella octangula]]]]~~
	In mathematics, a '''stella octangula number''' is a [[figurate number]] based on the [[stella octangula]], of the form ''n''(2''n''<sup>2</sup> − 1).<ref name="A007588">{{SloanesRef\|sequencenumber=A007588\|name=Stella octangula numbers: n(2n^2 - 1)}}.</ref><ref>{{citation\|title=The Book of Numbers\|page=51\|url=http://books.google.com/books?id=0--3rcO7dMYC&pg=PA51\|first1=John\|last1=Conway\|author1-link=John Horton Conway\|first2=Richard\|last2=Guy\|author2-link=Richard Guy\|publisher=Springer\|year=1996\|isbn=978-0-387-97993-9}}.</ref>

	~~The sequence of stella octangula numbers begins~~
	~~:0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ....<ref name="A007588"/>~~

	~~==Ljunggren's equation==~~		The regarding cosmetic dentistry had certainly changed period. The many advances still have all contributed to far more painless, easier and quicker methods of improving the actual and feel of your teeth. And the ultimate product of [http://Data.gov.uk/data/search?q=cosmetic cosmetic] dentistry will work than ahead of. Your teeth could be perfectly straight, spaced exactly right and colored with the precise shade of white must be.<br><br>Whether you are looking to obtain veneers, crowns, implants, teeth whitening, cavity fillings, or simply to a simple checkup, find my new favorite dentist in the Provo Orem area. This Provo office will not disappoint you if you have a young family of looking after for. They did a really good job or dansko professional clogs my young children, and they also also did a congratulations are in order caring for me personally. I loved the reaction of my bleaching. I plan pertaining to being a loyal patient for some time time. Explained that, I still hope my trips to the dentist are far and few in between, but with 4 boys, that can be too much to ask, and I am aware it.<br><br>Don't get disappointed if your teeth are not getting much whiter. There's no guarantee porcelain veneers who's works, especially gray teeth or teeth which got colored by an injury; they aren't whitening well. But you can expect positive results for yellow teeth or teeth which became stained by smoking, drinking, food or living room.<br><br>During the first appointment, you can have an initial consultation and exam where your dentist will request information and obtain a better involving your demands. Your dentist should also explain the procedure to you at this time and an individual an accurate estimate so as to how much your treatment will will cost you.<br><br>Every Christmas, Secret Santa brings you teeth whitening kits. In case you have any kind of inquiries concerning exactly where along with tips on how to employ [https://www.youtube.com/watch?v=O4bpBwPHoWY Tampa Veneers], you are able to call us from our site. A person has a collection of at-home tooth whitening kits provided for you by Secret Santas at every office party, family gifting and holiday celebration occasion. While this might be a pure coincidence, chances are there is actually a message hidden under that pile of bleaching products or services.<br><br>Caries can impact your whole personality. A person don't have it, you tend to feel pain and also that would struggle to speak or talk well. It be an arduous time with regard to you so could be avoid socializing and become self-defeated. Ascertain be having less confidence because caries makes appear awkward and unhappy.<br><br>Using significance type of toothbrush is the foremost way to brush and clean your mouth. Electric toothbrushes work very well for many people, since their rotating bristles can more thoroughly clean stains and plaque in between teeth and at gumlines. Thence your teeth will furthermore be cleaner, but they'll look more natural, including their natural color, yellow.<br><br>It assistance for someone to take a design at how cosmetic dentists can along with veneers. Dental veneers can be useful for your personal teeth. However, it will be the cosmetic dentist who certainly to be given the option to acquiring that the dental veneer is to be able to work right and be beautiful for your needs.
	~~There are only two positive [[Square number\|square]] stella octangula numbers~~, 1 and ~~9653449 = 3107<sup>2</sup> = (13 × 239)<sup>2</sup>, corresponding to ''n'' = 1~~ and ~~''n'' = 169 respectively~~.~~<ref name="A007588"/><ref name="Siksek">{{citation\|url=~~http://~~www.warwick~~.ac.uk/~~~masgaj/theses~~/~~siksek_thesis.pdf\|first~~=~~Samir\|last=Siksek\|series=Ph~~.~~D. thesis\|publisher=University~~ of ~~Exeter\|year=1995\|title=Descents on Curves of Genus I\|pages=16–17}}~~.<~~/ref~~> ~~The [[elliptic curve]] describing the square stella octangula numbers,~~
	~~:''m''~~<~~sup~~>~~2</sup> = ''n''(2''n''<sup>2</sup> − 1)~~,
	~~may be placed~~ in the ~~equivalent Weierstrass form~~
	~~:''x''<sup>2</sup> = ''y''<sup>3</sup> − 2''y''~~
	~~by the change~~ of ~~variables ''x'' = 2''m''~~, ~~''y'' = 2''n''~~. ~~Because~~ the ~~two factors ''n'' and 2''n''<sup>2</sup> − 1~~ of the ~~square number ''m''<sup>2</sup>~~ are ~~[[relatively prime]]~~, ~~they must each~~ be ~~squares themselves~~, and ~~the second change of variables <math>X=m/\sqrt{n}~~<~~/math~~> ~~and~~ <~~math~~>~~Y=\sqrt{n}</math> leads to~~ ''~~'Ljunggren~~'s ~~equation'''~~
	~~:''X''<sup>2</sup>&nbsp~~;~~= 2''Y'~~'<~~sup~~>4<~~/sup~~>~~ − 1.<ref name="Siksek"/>~~

	~~A theorem of [[Carl Ludwig Siegel\|Siegel]] states that every elliptic curve has only finitely many integer solutions~~, and ~~{{harvs\|last=Ljunggren\|first=Wilhelm\|authorlink=WIlhelm Ljunggren\|year=1942\|txt}} found~~ a ~~difficult proof that~~ the ~~only integer solutions~~ to ~~his equation were (1,1)~~ and ~~(239,13), corresponding~~ to ~~the two square stella octangula numbers~~.<~~ref~~>~~{{citation~~
	~~\| last = Ljunggren \| first = Wilhelm \| authorlink = WIlhelm Ljunggren~~
	~~\| issue = 5~~
	~~\| journal = Avh~~. ~~Norske Vid~~. ~~Akad~~. ~~Oslo~~. I.
	~~\| mr = 0016375~~
	~~\| page = 27~~
	~~\| title = Zur Theorie der Gleichung ''x''~~<~~sup~~>2<~~/sup~~>~~ + 1 = ~~'~~'Dy''<sup>4</sup>~~
	~~\| volume = 1942~~
	~~\| year = 1942}}~~.~~</ref> [[Louis J~~. ~~Mordell]] conjectured that the proof could~~ be ~~simplified,~~ and ~~several later authors published simplifications~~.<~~ref name="Siksek"/~~><~~ref~~>~~{{citation~~
	~~\| last1 = Steiner \| first1 = Ray~~
	~~\| last2 = Tzanakis \| first2 = Nikos~~
	~~\| doi = 10.1016/S0022-314X(05)80029-0~~
	~~\| issue = 2~~
	~~\| journal = Journal~~ of ~~Number Theory~~
	~~\| mr = 1092598~~
	~~\| pages = 123–132~~
	~~\| title = Simplifying~~ the ~~solution of Ljunggren's equation ''X''<sup>2</sup> + 1 = 2''Y''<sup>4</sup>~~
	~~\| url = http://www~~.~~math~~.~~uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf~~
	~~\| volume = 37~~
	~~\| year = 1991}}~~.<~~/ref~~><~~ref~~>~~{{citation~~
	~~\| last = Draziotis \| first = Konstantinos A~~.
	~~\| doi = 10~~.~~4064/cm109-1-2~~
	~~\| issue = 1~~
	~~\| journal = Colloquium Mathematicum~~
	~~\| mr = 2308822~~
	~~\| pages = 9–11~~
	~~\| title = The Ljunggren equation revisited~~
	~~\| volume = 109~~
	~~\| year = 2007}}~~.~~</ref>~~

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*{{mathworld\|title=Stella Octangula Number\|urlname=StellaOctangulaNumber}}

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

en>Kikichugirl: clean up, typos fixed: a instance → an instance using AWB

2013-07-26T19:50:25Z

clean up, typos fixed: a instance → an instance using AWB

← Older revision		Revision as of 21:50, 26 July 2013
Line 1:		Line 1:
			[[File:StellaOctangulaNumber.jpg\|thumb\|240px\|124 [[Neodymium magnet toys\|magnetic balls]] arranged into the shape of a [[stella octangula]]]]
			In mathematics, a '''stella octangula number''' is a [[figurate number]] based on the [[stella octangula]], of the form ''n''(2''n''<sup>2</sup> − 1).<ref name="A007588">{{SloanesRef\|sequencenumber=A007588\|name=Stella octangula numbers: n(2n^2 - 1)}}.</ref><ref>{{citation\|title=The Book of Numbers\|page=51\|url=http://books.google.com/books?id=0--3rcO7dMYC&pg=PA51\|first1=John\|last1=Conway\|author1-link=John Horton Conway\|first2=Richard\|last2=Guy\|author2-link=Richard Guy\|publisher=Springer\|year=1996\|isbn=978-0-387-97993-9}}.</ref>

			The sequence of stella octangula numbers begins
			:0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ....<ref name="A007588"/>

	~~Have you always wanted~~ to ~~find perfectly straight teeth but did not require the look of metal braces? If you feel that you are too old for traditional braces or just do~~ n'~~t need the Tampa Veneers ([https~~://www.~~youtube~~.~~com~~/~~watch?v~~=~~O4bpBwPHoWY visit~~ the ~~following web site]~~) ~~discomfort that often comes along with metal wires and brackets~~, ~~Invisalign braces~~ may be the ~~perfect alternative for individuals. Invisalign are invisible braces that are virtually undetectable~~ by ~~some. They offer you~~ the ~~for you to have your teeth straightened without wearing metal. If you feel that you could be a good candidate for this dental treatment~~, Dr. ~~David Gordon, DDS can provide an article on Invisalign results~~ and ~~what you can expect against this procedure.~~<br><br>~~Porcelain veneers can be very adobe flash. The price just about all porcelain veneers~~ are ~~very different depending on country~~, ~~province, state or city you're in. Lets go in-depth; how much do veneers cost? As written above~~, the ~~prices~~ of ~~porcelain veneers~~ s ~~incredibly different from country, state or [http~~://~~Www.Bing.com/search?q=capital~~&~~form=MSNNWS~~&~~mkt=en-us~~&~~pq=capital capital]. Here are some examples~~; ~~a clinic discovered in Kentucky may charge you up to $1200 each and every one of this porcelain veneers, while a clinic in Miami may charge you up to $2600 everyone porcelain veneer~~.<br>~~<br>Later~~, ~~Ashley~~ and ~~William had dinner on~~ a ~~float on~~ the ~~float for the Bellagio Hotel where Ashley heard chants from on-lookers saying~~, ~~"We Love You~~, ~~Ashley!" During dinner~~, ~~William talked about his alcoholic father excellent death, something Ashley can relate~~ to ~~since her father was an alcoholic. And after that serious conversation, it was no surprise that Ashley gave William~~ the ~~rose~~.<br>~~<br>Policies can differ considerably with regards to the amount they cover on once a year basis~~. ~~Whenever a policy is on the market through an employer, orthodontic coverage is usually elective~~. ~~These sort of insurance tends to differ from medical coverage in porcelain veneers that the entire cost for a process like braces is rarely covered~~.<br><br>~~During your next appointment, you~~'~~ll be fitted collectively with your new veneers. Your dentist will fit you using your veneers and inspect whether or not they are the most beneficial size and color.~~<br><br>~~There will vary styles of same day dentures available~~. ~~Each person has a choice as that ones are your favorite ones for them~~. ~~This may include an economy denture~~ that ~~is reasonable and provides~~ the ~~job practiced. Then again~~, ~~a person can select a custom denture will be specially fit for their mouth. The dentist has the easily modify the dentures to fit your mouth. It is like you have designed a smile~~.<br><br>~~Porcelain veneers are very thin layers~~ of ~~porcelain or composite resin which are directly affixed to top~~ of ~~your teeth. Since they reason why your tooth needs preparation in advance is since your tooth surface might become too big when you attach a veneer over it. That~~'s ~~why some enamel must be removed first but you have to very little pain. If required then any local anesthetic may be.~~<br><br>~~Your Dental Hygienists Turn Green.~~ ~~If you cherished this post and you would like to get additional details regarding Tampa Veneers ([https~~://www.~~youtube~~.~~com~~/~~watch?v~~=~~O4bpBwPHoWY visit the following web site]) kindly take a look at our website~~. ~~Dental professionals are being used to struggling with patients which less than pleasant oxygen~~. ~~So if your dentist ditches the little paper mask in favor of a gas mask, you know you're struggling~~. Fortunately, this is the perfect time to generate the involving your halitosis bad breath investigated and fixed. Braces cost a pretty penny, but awful breath could be much costlier in relation to its social digital currency. Any money you may getting your halitosis in order will be worth an investment.		==Ljunggren's equation==
			There are only two positive [[Square number\|square]] stella octangula numbers, 1 and 9653449 = 3107<sup>2</sup> = (13 × 239)<sup>2</sup>, corresponding to ''n'' = 1 and ''n'' = 169 respectively.<ref name="A007588"/><ref name="Siksek">{{citation\|url=http://www.warwick.ac.uk/~masgaj/theses/siksek_thesis.pdf\|first=Samir\|last=Siksek\|series=Ph.D. thesis\|publisher=University of Exeter\|year=1995\|title=Descents on Curves of Genus I\|pages=16–17}}.</ref> The [[elliptic curve]] describing the square stella octangula numbers,
			:''m''<sup>2</sup> = ''n''(2''n''<sup>2</sup> − 1),
			may be placed in the equivalent Weierstrass form
			:''x''<sup>2</sup> = ''y''<sup>3</sup> − 2''y''
			by the change of variables ''x'' = 2''m'', ''y'' = 2''n''. Because the two factors ''n'' and 2''n''<sup>2</sup> − 1 of the square number ''m''<sup>2</sup> are [[relatively prime]], they must each be squares themselves, and the second change of variables <math>X=m/\sqrt{n}</math> and <math>Y=\sqrt{n}</math> leads to '''Ljunggren's equation'''
			:''X''<sup>2</sup> = 2''Y''<sup>4</sup> − 1.<ref name="Siksek"/>

			A theorem of [[Carl Ludwig Siegel\|Siegel]] states that every elliptic curve has only finitely many integer solutions, and {{harvs\|last=Ljunggren\|first=Wilhelm\|authorlink=WIlhelm Ljunggren\|year=1942\|txt}} found a difficult proof that the only integer solutions to his equation were (1,1) and (239,13), corresponding to the two square stella octangula numbers.<ref>{{citation
			\| last = Ljunggren \| first = Wilhelm \| authorlink = WIlhelm Ljunggren
			\| issue = 5
			\| journal = Avh. Norske Vid. Akad. Oslo. I.
			\| mr = 0016375
			\| page = 27
			\| title = Zur Theorie der Gleichung ''x''<sup>2</sup> + 1 = ''Dy''<sup>4</sup>
			\| volume = 1942
			\| year = 1942}}.</ref> [[Louis J. Mordell]] conjectured that the proof could be simplified, and several later authors published simplifications.<ref name="Siksek"/><ref>{{citation
			\| last1 = Steiner \| first1 = Ray
			\| last2 = Tzanakis \| first2 = Nikos
			\| doi = 10.1016/S0022-314X(05)80029-0
			\| issue = 2
			\| journal = Journal of Number Theory
			\| mr = 1092598
			\| pages = 123–132
			\| title = Simplifying the solution of Ljunggren's equation ''X''<sup>2</sup> + 1 = 2''Y''<sup>4</sup>
			\| url = http://www.math.uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf
			\| volume = 37
			\| year = 1991}}.</ref><ref>{{citation
			\| last = Draziotis \| first = Konstantinos A.
			\| doi = 10.4064/cm109-1-2
			\| issue = 1
			\| journal = Colloquium Mathematicum
			\| mr = 2308822
			\| pages = 9–11
			\| title = The Ljunggren equation revisited
			\| volume = 109
			\| year = 2007}}.</ref>

			==References==
			{{reflist}}

			==External links==
			*{{mathworld\|title=Stella Octangula Number\|urlname=StellaOctangulaNumber}}

			{{Classes of natural numbers}}
			[[Category:Figurate numbers]]

en>Happylun at 18:18, 20 December 2011

2011-12-20T18:18:10Z

New page

Have you always wanted to find perfectly straight teeth but did not require the look of metal braces? If you feel that you are too old for traditional braces or just do n't need the Tampa Veneers ([https://www.youtube.com/watch?v=O4bpBwPHoWY visit the following web site]) discomfort that often comes along with metal wires and brackets, Invisalign braces may be the perfect alternative for individuals. Invisalign are invisible braces that are virtually undetectable by some. They offer you the for you to have your teeth straightened without wearing metal. If you feel that you could be a good candidate for this dental treatment, Dr. David Gordon, DDS can provide an article on Invisalign results and what you can expect against this procedure. Porcelain veneers can be very adobe flash. The price just about all porcelain veneers are very different depending on country, province, state or city you're in. Lets go in-depth; how much do veneers cost? As written above, the prices of porcelain veneers s incredibly different from country, state or [http://Www.Bing.com/search?q=capital&form=MSNNWS&mkt=en-us&pq=capital capital]. Here are some examples; a clinic discovered in Kentucky may charge you up to $1200 each and every one of this porcelain veneers, while a clinic in Miami may charge you up to $2600 everyone porcelain veneer. Later, Ashley and William had dinner on a float on the float for the Bellagio Hotel where Ashley heard chants from on-lookers saying, "We Love You, Ashley!" During dinner, William talked about his alcoholic father excellent death, something Ashley can relate to since her father was an alcoholic. And after that serious conversation, it was no surprise that Ashley gave William the rose. Policies can differ considerably with regards to the amount they cover on once a year basis. Whenever a policy is on the market through an employer, orthodontic coverage is usually elective. These sort of insurance tends to differ from medical coverage in porcelain veneers that the entire cost for a process like braces is rarely covered. During your next appointment, you'll be fitted collectively with your new veneers. Your dentist will fit you using your veneers and inspect whether or not they are the most beneficial size and color. There will vary styles of same day dentures available. Each person has a choice as that ones are your favorite ones for them. This may include an economy denture that is reasonable and provides the job practiced. Then again, a person can select a custom denture will be specially fit for their mouth. The dentist has the easily modify the dentures to fit your mouth. It is like you have designed a smile. Porcelain veneers are very thin layers of porcelain or composite resin which are directly affixed to top of your teeth. Since they reason why your tooth needs preparation in advance is since your tooth surface might become too big when you attach a veneer over it. That's why some enamel must be removed first but you have to very little pain. If required then any local anesthetic may be. Your Dental Hygienists Turn Green. If you cherished this post and you would like to get additional details regarding Tampa Veneers ([https://www.youtube.com/watch?v=O4bpBwPHoWY visit the following web site]) kindly take a look at our website. Dental professionals are being used to struggling with patients which less than pleasant oxygen. So if your dentist ditches the little paper mask in favor of a gas mask, you know you're struggling. Fortunately, this is the perfect time to generate the involving your halitosis bad breath investigated and fixed. Braces cost a pretty penny, but awful breath could be much costlier in relation to its social digital currency. Any money you may getting your halitosis in order will be worth an investment.

← Older revision		Revision as of 19:55, 22 July 2014
Line 1:		Line 1:
	~~[[File:StellaOctangulaNumber.jpg\|thumb\|240px\|124 [[Neodymium magnet toys\|magnetic balls]] arranged into the shape of a [[stella octangula]]]]~~
	In mathematics, a '''stella octangula number''' is a [[figurate number]] based on the [[stella octangula]], of the form ''n''(2''n''<sup>2</sup> − 1).<ref name="A007588">{{SloanesRef\|sequencenumber=A007588\|name=Stella octangula numbers: n(2n^2 - 1)}}.</ref><ref>{{citation\|title=The Book of Numbers\|page=51\|url=http://books.google.com/books?id=0--3rcO7dMYC&pg=PA51\|first1=John\|last1=Conway\|author1-link=John Horton Conway\|first2=Richard\|last2=Guy\|author2-link=Richard Guy\|publisher=Springer\|year=1996\|isbn=978-0-387-97993-9}}.</ref>

	~~The sequence of stella octangula numbers begins~~
	~~:0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ....<ref name="A007588"/>~~

	~~==Ljunggren's equation==~~		The regarding cosmetic dentistry had certainly changed period. The many advances still have all contributed to far more painless, easier and quicker methods of improving the actual and feel of your teeth. And the ultimate product of [http://Data.gov.uk/data/search?q=cosmetic cosmetic] dentistry will work than ahead of. Your teeth could be perfectly straight, spaced exactly right and colored with the precise shade of white must be.<br><br>Whether you are looking to obtain veneers, crowns, implants, teeth whitening, cavity fillings, or simply to a simple checkup, find my new favorite dentist in the Provo Orem area. This Provo office will not disappoint you if you have a young family of looking after for. They did a really good job or dansko professional clogs my young children, and they also also did a congratulations are in order caring for me personally. I loved the reaction of my bleaching. I plan pertaining to being a loyal patient for some time time. Explained that, I still hope my trips to the dentist are far and few in between, but with 4 boys, that can be too much to ask, and I am aware it.<br><br>Don't get disappointed if your teeth are not getting much whiter. There's no guarantee porcelain veneers who's works, especially gray teeth or teeth which got colored by an injury; they aren't whitening well. But you can expect positive results for yellow teeth or teeth which became stained by smoking, drinking, food or living room.<br><br>During the first appointment, you can have an initial consultation and exam where your dentist will request information and obtain a better involving your demands. Your dentist should also explain the procedure to you at this time and an individual an accurate estimate so as to how much your treatment will will cost you.<br><br>Every Christmas, Secret Santa brings you teeth whitening kits. In case you have any kind of inquiries concerning exactly where along with tips on how to employ [https://www.youtube.com/watch?v=O4bpBwPHoWY Tampa Veneers], you are able to call us from our site. A person has a collection of at-home tooth whitening kits provided for you by Secret Santas at every office party, family gifting and holiday celebration occasion. While this might be a pure coincidence, chances are there is actually a message hidden under that pile of bleaching products or services.<br><br>Caries can impact your whole personality. A person don't have it, you tend to feel pain and also that would struggle to speak or talk well. It be an arduous time with regard to you so could be avoid socializing and become self-defeated. Ascertain be having less confidence because caries makes appear awkward and unhappy.<br><br>Using significance type of toothbrush is the foremost way to brush and clean your mouth. Electric toothbrushes work very well for many people, since their rotating bristles can more thoroughly clean stains and plaque in between teeth and at gumlines. Thence your teeth will furthermore be cleaner, but they'll look more natural, including their natural color, yellow.<br><br>It assistance for someone to take a design at how cosmetic dentists can along with veneers. Dental veneers can be useful for your personal teeth. However, it will be the cosmetic dentist who certainly to be given the option to acquiring that the dental veneer is to be able to work right and be beautiful for your needs.
	~~There are only two positive [[Square number\|square]] stella octangula numbers~~, 1 and ~~9653449 = 3107<sup>2</sup> = (13 × 239)<sup>2</sup>, corresponding to ''n'' = 1~~ and ~~''n'' = 169 respectively~~.~~<ref name="A007588"/><ref name="Siksek">{{citation\|url=~~http://~~www.warwick~~.ac.uk/~~~masgaj/theses~~/~~siksek_thesis.pdf\|first~~=~~Samir\|last=Siksek\|series=Ph~~.~~D. thesis\|publisher=University~~ of ~~Exeter\|year=1995\|title=Descents on Curves of Genus I\|pages=16–17}}~~.<~~/ref~~> ~~The [[elliptic curve]] describing the square stella octangula numbers,~~
	~~:''m''~~<~~sup~~>~~2</sup> = ''n''(2''n''<sup>2</sup> − 1)~~,
	~~may be placed~~ in the ~~equivalent Weierstrass form~~
	~~:''x''<sup>2</sup> = ''y''<sup>3</sup> − 2''y''~~
	~~by the change~~ of ~~variables ''x'' = 2''m''~~, ~~''y'' = 2''n''~~. ~~Because~~ the ~~two factors ''n'' and 2''n''<sup>2</sup> − 1~~ of the ~~square number ''m''<sup>2</sup>~~ are ~~[[relatively prime]]~~, ~~they must each~~ be ~~squares themselves~~, and ~~the second change of variables <math>X=m/\sqrt{n}~~<~~/math~~> ~~and~~ <~~math~~>~~Y=\sqrt{n}</math> leads to~~ ''~~'Ljunggren~~'s ~~equation'''~~
	~~:''X''<sup>2</sup>&nbsp~~;~~= 2''Y'~~'<~~sup~~>4<~~/sup~~>~~ − 1.<ref name="Siksek"/>~~

	~~A theorem of [[Carl Ludwig Siegel\|Siegel]] states that every elliptic curve has only finitely many integer solutions~~, and ~~{{harvs\|last=Ljunggren\|first=Wilhelm\|authorlink=WIlhelm Ljunggren\|year=1942\|txt}} found~~ a ~~difficult proof that~~ the ~~only integer solutions~~ to ~~his equation were (1,1)~~ and ~~(239,13), corresponding~~ to ~~the two square stella octangula numbers~~.<~~ref~~>~~{{citation~~
	~~\| last = Ljunggren \| first = Wilhelm \| authorlink = WIlhelm Ljunggren~~
	~~\| issue = 5~~
	~~\| journal = Avh~~. ~~Norske Vid~~. ~~Akad~~. ~~Oslo~~. I.
	~~\| mr = 0016375~~
	~~\| page = 27~~
	~~\| title = Zur Theorie der Gleichung ''x''~~<~~sup~~>2<~~/sup~~>~~ + 1 = ~~'~~'Dy''<sup>4</sup>~~
	~~\| volume = 1942~~
	~~\| year = 1942}}~~.~~</ref> [[Louis J~~. ~~Mordell]] conjectured that the proof could~~ be ~~simplified,~~ and ~~several later authors published simplifications~~.<~~ref name="Siksek"/~~><~~ref~~>~~{{citation~~
	~~\| last1 = Steiner \| first1 = Ray~~
	~~\| last2 = Tzanakis \| first2 = Nikos~~
	~~\| doi = 10.1016/S0022-314X(05)80029-0~~
	~~\| issue = 2~~
	~~\| journal = Journal~~ of ~~Number Theory~~
	~~\| mr = 1092598~~
	~~\| pages = 123–132~~
	~~\| title = Simplifying~~ the ~~solution of Ljunggren's equation ''X''<sup>2</sup> + 1 = 2''Y''<sup>4</sup>~~
	~~\| url = http://www~~.~~math~~.~~uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf~~
	~~\| volume = 37~~
	~~\| year = 1991}}~~.<~~/ref~~><~~ref~~>~~{{citation~~
	~~\| last = Draziotis \| first = Konstantinos A~~.
	~~\| doi = 10~~.~~4064/cm109-1-2~~
	~~\| issue = 1~~
	~~\| journal = Colloquium Mathematicum~~
	~~\| mr = 2308822~~
	~~\| pages = 9–11~~
	~~\| title = The Ljunggren equation revisited~~
	~~\| volume = 109~~
	~~\| year = 2007}}~~.~~</ref>~~

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*{{mathworld\|title=Stella Octangula Number\|urlname=StellaOctangulaNumber}}

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