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,

${\displaystyle m^{2}=n(2n^{2}-1)}$

may be placed in the equivalent Weierstrass form

${\displaystyle x^{2}=y^{3}-2y}$

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 ${\displaystyle X=m/{\sqrt {n))}$ and ${\displaystyle Y={\sqrt {n))}$ leads to Ljunggren's equation

${\displaystyle X^{2}=2Y^{4}-1}$[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]