Stella octangula number
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
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,
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.[4] Louis J. Mordell conjectured that the proof could be simplified, and several later authors published simplifications.[3][5][6]
References
- 1 2 3 "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.
- 1 2 3 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.