On the equation x² - 2 z⁶ = y^p.
by
Imin Chen
Simon Fraser University

The modular method has been successfully applied to tackle a number of classes of ternary diophantine equations of the form A x^a + B y^b = C z^c. Of interest sometimes are equations obtained by setting one of the variables x, y, z to 1. The equation x² - 2 = y^p is an example, but it has resisted attempts so far because the solution (x, y) = (±1, -1) is present for every p and the associated elliptic curves over Q from the modular method do not have complex multiplication. By regarding this equation as a special case of x² - 2 z⁶ = y^p, we show that it is possible to associate to a solution a Q-curve completely defined over Q(√2, √3). The solution (±1, -1) now luckily corresponds to an elliptic curve with complex multiplication by the order of discriminant -24 and the modular method using Q-curves then can be applied to obtain some partial results.

Date received: May 14, 2008