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 begins

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ....[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,

m2 = n(2n2 − 1),

may be placed in the equivalent Weierstrass form

x2 = y3 − 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

X2 = 2Y4 − 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. ↑ 1.0 1.1 1.2 "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. ↑ 3.0 3.1 3.2 Siksek, Samir (1995), Descents on Curves of Genus I, 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", 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 .

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