Diophantine equations in positive integer solutions

Question

I want to know the solution of the equation $x^3$ + $y^3$ = $31z^3$ in integers. I know the fundamental solution ($137, -65$, $42$), but want to have all the values positive. I know also that there is an aritmetic procedure (doubling in the group) to obtain further solutions from the fundamental one, but I do not know the details of this procedure. Could you explain these details, and how to get the first positive solution, in this case and in other cases with fundamental non-positive solutions? Thanks in advance

Answer 1

I don`t know a lot about this method, but it is part of elliptic curves. The idea is to write the equation in two variables by making a change of variables, r = x/z and r = y/z. Then if you have two rational points on the new curve, you can draw a line through them and the line will intersect the curve at a third point which represents the sum of the points, which is also rational. If you only have one point, you can draw the tangent at that point and the intersection will be a rational point (this is 'doubling' the given point).

In your example you have the curve

$r^3$ + $s^3$ = 31

and a rational point r = 137/42, s = -65/42. If you draw the tangent at this point and figure its intersection with the curve, you`ll get another point

r = 277028111/119531076, s = 316425265/119531076

which corresponds to the solution of the original equation

x= 277028111, y = 316425265, z = 119531076.