Stella octangula number

Ljunggren's equation

There are only two positive square stella octangula numbers, 1 and , corresponding to and respectively. The elliptic curve describing the square stella octangula numbers, :m^2 = n (2n^2 - 1) may be placed in the equivalent Weierstrass form :x^2 = y^3 - 2y by the change of variables , . Because the two factors n and 2n − 1 of the square number m 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 :X^2 = 2Y^4 - 1

A theorem of Siegel states that every elliptic curve has only finitely many integer solutions, and 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.{{citation

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.{{citation

References

References

1. {{Cite OEIS
2. (1996). "The Book of Numbers". Springer.
3. Siksek, Samir. (1995). "Descents on Curves of Genus I". University of Exeter.