stella octangula number

{{Short description|Figurate number based on the stella octangula}}

File:StellaOctangulaNumber.jpg 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(2n² − 1)}}.{{Cite OEIS|sequencenumber=A007588|name=Stella octangula numbers: n*(2*n^2 - 1)|mode=cs2}}.{{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}}.

The sequence of stella octangula numbers is

:0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... {{OEIS|A007588}}

Only two of these numbers are square.

Ljunggren's equation

There are only two positive square stella octangula numbers, {{math|1}} and {{math|1=9653449 = 3107{{sup|2}} = (13 × 239){{sup|2}}}}, corresponding to {{math|1=n = 1}} and {{math|1=n = 169}} respectively.{{citation|url=https://homepages.warwick.ac.uk/staff/S.Siksek/papers/phdnew.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}} The elliptic curve describing the square stella octangula numbers,

: $m^2 = n (2n^2 - 1)$

may be placed in the equivalent Weierstrass form

: $x^2 = y^3 - 2y$

by the change of variables {{math|1=x = 2m}}, {{math|1=y = 2n}}. Because the two factors {{mvar|n}} and {{math|2n{{sup|2}} − 1}} of the square number {{math|m{{sup|2}}}} are relatively prime, they must each be squares themselves, and the second change of variables $X=m/\sqrt{n}$ and $Y=\sqrt{n}$ leads to Ljunggren's equation

: $X^2 = 2Y^4 - 1$

A theorem of 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.{{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 x² + 1 = Dy⁴

| volume = 1942

| year = 1942}}. Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.{{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=X² + 1 = 2Y⁴}}

| url = http://www.math.uoc.gr/~tzanakis/Papers/LjunggrenEq.pdf

| volume = 37

| year = 1991| doi-access = free

}}.{{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| doi-access = free

}}.

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.{{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.de/journals/AMI/2009/ami2009-bremner-hoibakk-lukkassen.pdf

| volume = 36

| year = 2009}}.

References

External links

{{mathworld|title=Stella Octangula Number|urlname=StellaOctangulaNumber|mode=cs2}}

Category:Figurate numbers