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(2n² − 1).^[1][2]

The sequence of stella octangula numbers is

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

Ljunggren's equation

There are only two positive square stella octangula numbers, 1 and 9653449 = 3107² = (13 × 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 = 2m, y = 2n. Because the two factors n and 2n² − 1 of the square number m² 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]

References

1. "Sloane's 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.
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 X² + 1 = 2Y⁴" (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.

External links

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