*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.

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