A Variant on Waring's Problem



Legendre showed that every positive integer can be written as the sum of four squares in 1770. Subsequently, in answer to a question of Waring, it was shown that every positive integer is the sum of 9 (non-negative) cubes, the sum of 19 fourth powers, etc.

What if we extend this investigation to rational numbers? It is not difficult to see that Legendre's result actually shows that every positive rational number is the sum of four rational squares and that this number cannot be decreased. Passing to the case of cubes, we claim that every positive rational number is the sum of three non-negative rational cubes.

Given a rational number, r, we are looking for rational solutions to x^3 + y^3 + z^3 = r. Clearing denominators, we are looking for integer solutions to a^3 + b^3 + c^3 = r d^3. Dividing by c^3 and letting s=a/c, t=b/c, and u=d/c we are looking for rational solutions to s^3 + t^3 + 1 = r u^3. But this has the trivial solution s= -1, t=0, u=0. If we take an arbitrary line through this point (note that there is a two-parameter family of these) and intersect it with our cubic surface, in general we will not get rational points but points whose coordinates lie in the field Q(Sqrt[k]) = {x+y Sqrt[k]| x and y are rational}, where k is some rational number. If we take one of these points, find the equation of the tangent plane to the cubic surace at this point and intersect the surface with this plane, we will get a singular cubic curve with coefficients in Q(Sqrt[k]).

Now applying the techniques of section 2, we can construct a one-parameter family of points on this singular cubic with coordinates in Q(Sqrt[k]). Take one of them and denote it by P. Let f denote the function from Q(Sqrt[k])^3 to itself given by taking each coordinate x + y Sqrt[k] to its conjugate, x - y Sqrt[k]. Consider the line through the points P and f(P). This line will intersect our cubic surface in a third point, which we claim has rational coordinates. To see this, note that the set of points on our surface having coordinates in Q(Sqrt[k]) is carried to itself by f and the line containing P and f(P) is carried to itself by f. Therefore the intersection of the line and the cubic surface is carried to itself by f. But this means that the third point of intersection must be sent to itself by f and the only way this can occur is if the coordinates of that point are rational.

Finally we claim that we have enough parameters in the solution above to allow us to construct positive rational solutions to our equation.

Next: Two Challenges
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