*A Variant on Waring's Problem*

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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

e-mail Les Reid