- #1

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let y = x +α and z = x + β

now by simple substitution we get the following cubic equation:

x^3 + 3(α-β)x^2 + 3(α^2 - β^2)x + α^3 - β^3 = 0

So all that is left to prove is that the above cubic equation can never have integer solutions.

Here we can try to use the work done by Konstantine Zelator here

http://arxiv.org/ftp/arxiv/papers/1110/1110.6110.pdf

where he gives conditions for a cubic polynomial to have integer roots.

The problem is that I am having a bit of a problem applying his theorem to the case above. Any help will be greatly appreciated.

Another question that comes to mind is: Can this method be generalized to the case with exponent greater than 3?