stella octangula number

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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 term can mean * polyg ...
based on the , 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 Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
.

# Ljunggren's equation

There are only two positive
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
stella octangula numbers, and , corresponding to and respectively.. The
elliptic curve In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
describing the square stella octangula numbers, :$m^2 = n \left(2n^2 - 1\right)$ 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 mathematics, 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 equivale ...
, 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 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 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..

# External links

* {{Classes of natural numbers Figurate numbers