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Problem:

Given the power diophantine equation $$\sum_{i=1}^{n} x_i^{c}= \sum_{j=1}^{n} y_j^{c} \tag{1}$$

where $(1)$ follows three conditions:

  1. Variable $x_i \neq y_j$ for any indexes $i, j\leq n$.

  2. Variable $x_i \neq x_k$ for any indexes $i, k\leq n$.

  3. Constant exponent $c\geq 2$.

Then, are there infinite solutions of $(1)$ for a fixed $c$ and a variable $n$ ?

Query: Is there anything related to the above problem in the existing literature?

Please inform me if it exists in any literature, book, note or web page.

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    Hard question take a look at "Arithmetics" from Hindry – 2017-02-19
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    @Maman , r u referring to Waring's problem ? – 2017-02-19
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    This teacher is a lover of diophantine equations and solutions of polynomial equations. Find his book it could be very interesting ! I give you the complete reference : http://www.springer.com/us/book/9781447121305 – 2017-02-19
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    This is called ***equal sums of like powers***. In particular, your post is the _balanced_ case where each side has the same number of terms. A google search should give many hits. – 2017-02-20
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    https://en.m.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture – 2017-02-20
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    https://en.m.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem – 2017-02-20

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