In the previous post, it was discussed that an initial non-trivial integer solution to the *diagonal quartic surface*,

apparently does not help in determining if the equation has an infinite number of primitive integer solutions. However, it is different for its *quadratic* and *cubic* counterparts.

** Theorem 1**: “

*In general, given one non-trivial solution to the quadratic*,

*then an infinite more can be found*.”

** Theorem 2**: “

*Likewise, given one non-trivial solution to the cubic*,

*then an infinite more can be found*.”

Proof of* Theorem 1 *(*Theorem 2* will be discussed in the next post):

We’ll start with *n* = 3,4 and prove the rest by induction. There is the identity,

where,

and,

Thus, if one has an initial solution , then the RHS of the identity becomes zero, and one gets a parametrization in the for three free variables . An example is given in this page, form 2b. For *n* = 4, it is just a generalization,

where,

and,

The pattern is easily seen for *n* = 5,6, *ad infinitum*. If only it was that easy for the quartic case.