Stella octangula number
The sequence of stella octangula numbers is
There are only two positive square stella octangula numbers, 1 and 9653449 = 31072 = (13 × 239)2, corresponding to n = 1 and n = 169 respectively. The elliptic curve describing the square stella octangula numbers,
may be placed in the equivalent Weierstrass form
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 and leads to Ljunggren's equation
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. Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.
- "Sloane's 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.
- 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 X2 + 1 = 2Y4" (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.