Stella octangula number

124 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 $n (2 n 2 - 1)$ .^[1]^[2]

The sequence of stella octangula numbers is

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... (sequence A007588 in the OEIS)^[1]

Only two of these numbers are square.

Ljunggren's equation

There are only two positive square stella octangula numbers, $1$ and $9653449 = 3107 2 = (13 \times 239) 2$ , corresponding to $n = 1$ and $n = 169$ respectively.^[1]^[3] 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 $x = 2 m$ , $y = 2 n$ . Because the two factors $n$ and $2 n 2 - 1$ of the square number $m 2$ 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

^[3]

A theorem of Siegel states that every elliptic curve has only finitely many integer solutions, and Wilhelm Ljunggren (1942) 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.^[4] Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.^[3]^[5]^[6]

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.^[7]

References

^ ^a ^b ^c Sloane, N. J. A. (ed.), "Sequence A007588 (Stella octangula numbers: n*(2*n^2 - 1))", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation.
^ Conway, John; Guy, Richard (1996), The Book of Numbers, Springer, p. 51, ISBN 978-0-387-97993-9.
^ ^a ^b ^c Siksek, Samir (1995), Descents on Curves of Genus I (PDF), Ph.D. thesis, University of Exeter, pp. 16–17
^ Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x² + 1 = Dy⁴", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27, MR 0016375.
^ Steiner, Ray; Tzanakis, Nikos (1991), "Simplifying the solution of Ljunggren's equation $X 2 + 1 = 2 Y 4$ " (PDF), Journal of Number Theory, 37 (2): 123–132, doi:10.1016/S0022-314X(05)80029-0, MR 1092598.
^ Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum, 109 (1): 9–11, doi:10.4064/cm109-1-2, MR 2308822.
^ Bremner, A.; Høibakk, R.; Lukkassen, D. (2009), "Crossed ladders and Euler's quartic" (PDF), Annales Mathematicae et Informaticae, 36: 29–41, MR 2580898.

External links

Weisstein, Eric W., "Stella Octangula Number", MathWorld

[A007588-1] Sloane, N. J. A. (ed.), "Sequence A007588 (Stella octangula numbers: n*(2*n^2 - 1))", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation.

[2] Conway, John; Guy, Richard (1996), The Book of Numbers, Springer, p. 51, ISBN 978-0-387-97993-9.

[Siksek-3] Siksek, Samir (1995), Descents on Curves of Genus I (PDF), Ph.D. thesis, University of Exeter, pp. 16–17

[4] Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x² + 1 = Dy⁴", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27, MR 0016375.

[5] Steiner, Ray; Tzanakis, Nikos (1991), "Simplifying the solution of Ljunggren's equation $X 2 + 1 = 2 Y 4$ " (PDF), Journal of Number Theory, 37 (2): 123–132, doi:10.1016/S0022-314X(05)80029-0, MR 1092598.

[6] Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum, 109 (1): 9–11, doi:10.4064/cm109-1-2, MR 2308822.

[7] Bremner, A.; Høibakk, R.; Lukkassen, D. (2009), "Crossed ladders and Euler's quartic" (PDF), Annales Mathematicae et Informaticae, 36: 29–41, MR 2580898.

[1]

[2]

[3]

[4]

[5]

[6]

[7]