Stella octangula number

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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 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 = 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

[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

  1. ^ 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. .
  2. ^ Conway, John; Guy, Richard (1996), The Book of Numbers, Springer, p. 51, ISBN 978-0-387-97993-9 .
  3. ^ a b c Siksek, Samir (1995), Descents on Curves of Genus I (PDF), Ph.D. thesis, University of Exeter, pp. 16–17 [permanent dead link].
  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 .
  7. ^ 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

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