# urbane diction art

quadtree voronoi secant

We want to reformulate Mordell's theorem in a way which has great aesthetic and technical advantages.In passing, we will mention that there is no known method to determine in a finite number of steps whether a given rational cubic has a rational point. There is no analogue of Hasse's theorem for cubics. That question is still open, and it is a very important question. The idea of looking modulo

mfor all integersmis not sufficient. Selmer gave the example`3X`

^{3}+ 4Y^{3}+ 5Z^{3}= 0.This is a cubic, and Selmer shows by an ingenious argument that it has no integer solutions other than

`(0, 0, 0)`

. However, one can check that for every integerm, the congruence`3X`

^{3}+ 4Y^{3}+ 5Z^{3}≡ 0 (modm)has a solution in integers with no common factor. So for general cubics, the existence of a solution modulo

mfor allmdoes not ensure that a rational solution exists. We will leave this difficult problem aside, and assume that we have a cubic which has a rational pointO.