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I know that Pythagorean triples have been parameterized, I also know that Andrew Wiles has proved that there are no distinct integer solutions for $ a^n + b^n = c^n$, when $ n \ge 3 $.

However we may then naturally ask, can cubic quartuples be parameterized? How about quartic quintuples?

In particular I have been able to solve for an arithmetic sequence of cubic quartuples with a common difference of 1 by solving the equation: $ x^3 = (x-1)^3 + (x-2)^3 + (x-3)^3 $.

Spoiler alert!!!

The solution is $x=6$

While I acknowledge that this result is too simple and trivial to be originally solve by myself. This sets me up for my big questions:

1) What is the general name for Diophantine equations of the form: $x^n=\sum_{i=1}^{n} (x-a_i)^n $ where $ x > |x-a_i|\forall a_i$, and $ x, a_i \in \mathbb{Z}$? For the case where $n=2$ this is obviously the pythagorean triple. What is this called in general?

2) Have any or all of these Diophantine equations been parameterized? I have done google searches on "parameterization of cubic quartuples", "parameterization of "quartic quintuples" and do not think that what I have found has been particularly relevent. This is why I ask.

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    $\begingroup$ Some good things to Google for are "equal sums of like powers" and "diagonal cubic/quartic" etc. In particular Euler made some (false!) conjectures about the sort of equation you are interested in and you can read a little about it at en.wikipedia.org/wiki/Euler's_sum_of_powers_conjecture a general formula for the cubic case is given there. I think there is some material on this sort of equation in Hardy and Wright's book but I might be misremembering. $\endgroup$ – Alex J Best Jul 12 '14 at 1:35
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You will get a good start on a very large subject with this Wikipedia article, which makes mention of some parametrizations. Titus Piezas has collected many diverse parametrizations of the general kind you are asking about.

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