In mathematics, a stella octangula number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...

based on the

stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicte ...

, of the form .. The sequence of stella octangula numbers is :0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... Only two of these numbers are

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

Ljunggren's equation

There are only two positive

stella octangula numbers, and , corresponding to and respectively. The

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

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 , . Because the two factors and of the square number are

relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

, they must each be squares themselves, and the second change of variables

X=m/\sqrt

and

Y=\sqrt

leads to Ljunggren's equation :

X^2 = 2Y^4 - 1

A theorem of

Siegel Siegel (also Segal, Segali or Segel), is a Germans, German and Ashkenazi Jewish surname. Alternate spellings include Sigel, Sigl, Siegl, and others. It can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed ...

states that every elliptic curve has only finitely many integer solutions, and found a difficult proof that the only integer solutions to his equation were and , corresponding to the two square stella octangula numbers. Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.

Additional applications

The stella octangula numbers arise in a parametric family of instances to the

crossed ladders problem The crossed ladders problem is a puzzle of unknown origin that has appeared in various publications and regularly reappears in Web pages and Usenet discussions. The problem Two ladders of lengths ''a'' and ''b'' lie oppositely across an alley, a ...

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..

References

External links

* {{Classes of natural numbers Figurate numbers