Theorem 2: “Likewise, given one non-trivial solution to the cubic,
then an infinite more can be found.”
Proof: As before, we will start with a particular example and derive the rest by induction. The following is identically true,
where,
Thus, if one has initial {} such that the RHS is zero, this leads to a second, the {}. By iteration, these can be used to generate a third, and so on. The reason why {} have a common factor will be clear in a moment.
This is by A. Desboves, but it is not hard to generalize it to n cubes. The basis is the identity,
If the first factor of the RHS, , can be expressed as a sum of n cubes,
it is a simple matter of distributing the second factor of the RHS among the n cubes so that (1) assumes the form,
which explains the common factor of {} in the 4-cube identity. Hence from {}, we get new solutions {} leading to further solutions, ad infinitum.
There doesn’t seem to be any example of a diagonal quartic surface,
which has been proven to have only one non-trivial and primitive integer solution. Based upon the quadratic and cubic cases, it is tempting to speculate that if it has one, then there may be in fact an infinity.