View source for Stella octangula number

[[File:StellaOctangulaNumber.jpg|thumb|upright=1.35|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 {{math|''n''(2''n''<sup>2</sup> &minus; 1)}}.<ref name="A007588">{{Cite OEIS|sequencenumber=A007588|name=Stella octangula numbers: n*(2*n^2 - 1)}}.</ref><ref>{{citation|title=The Book of Numbers|page=51|url=https://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 K. Guy|publisher=Springer|year=1996|isbn=978-0-387-97993-9}}.</ref>

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

Only two of these numbers are [[square number|square]].

==Ljunggren's equation==
There are only two positive [[Square number|square]] stella octangula numbers, {{math|1}} and {{math|1=9653449 = 3107{{sup|2}} = (13 &times; 239){{sup|2}}}}, corresponding to {{math|1=''n'' = 1}} and {{math|1=''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}}{{dead link|date=March 2018 |bot=InternetArchiveBot |fix-attempted=yes }}.</ref> The [[elliptic curve]] describing the square stella octangula numbers,
:<math>m^2 = n (2n^2 - 1)</math>
may be placed in the equivalent Weierstrass form
:<math>x^2 = y^3 - 2y</math>
by the change of variables {{math|1=''x'' = 2''m''}}, {{math|1=''y'' = 2''n''}}. Because the two factors {{mvar|n}} and {{math|2''n''{{sup|2}} &minus; 1}} of the square number {{math|''m''{{sup|2}}}} 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'''
:<math>X^2 = 2Y^4 - 1</math><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 {{math|(1,1)}} and {{math|(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>&nbsp;+&nbsp;1&nbsp;=&nbsp;''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 {{math|1=''X''<sup>2</sup>&nbsp;+&nbsp;1&nbsp;=&nbsp;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>

==Additional applications==
The stella octangula numbers arise in a parametric family of instances to the [[crossed ladders problem]] in which the lengths and heights of the ladders and the height of their crossing point are all integers. In these instances, the ratio between the heights of the two ladders is a stella octangula number.<ref>{{citation
 | last1 = Bremner | first1 = A.
 | last2 = Høibakk | first2 = R.
 | last3 = Lukkassen | first3 = D.
 | journal = Annales Mathematicae et Informaticae
 | mr = 2580898
 | pages = 29–41
 | title = Crossed ladders and Euler's quartic
 | url = http://www.emis.ams.org/journals/AMI/2009/ami2009-bremner-hoibakk-lukkassen.pdf
 | volume = 36
 | year = 2009}}.</ref>

==References==
{{reflist}}

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

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