Are there any integral solutions of the Diophantine equation

other than or such that ?

Solution:

Let

so that , or equivalently

……….. (1)

since (1) it is equivalent to

Hence an infinite class of non-trivial solutions is given by

I don’t know whether or not there are any other solutions.

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Like these?, format {x, y, z}

{-27,-8,-1}

{-27,-1,-8}

{-8,-27,1}

{-8,1,-27}

{-6,-48,6}

{-6,6,-48}

{-5,-40,5}

{-5,5,-40}

{-4,-32,4}

{-4,4,-32}

{-3,-24,3}

{-3,3,-24}

{-2,-16,2}

{-2,2,-16}

{-1,-27,8}

{-1,-8,1}

{-1,1,-8}

{-1,8,-27}

{1,-8,27}

{1,-1,8}

{1,8,-1}

{1,27,-8}

{2,-2,16}

{2,16,-2}

{3,-3,24}

{3,24,-3}

{4,-4,32}

{4,32,-4}

{5,-5,40}

{5,40,-5}

{6,-6,48}

{6,48,-6}

{8,-1,27}

{8,27,-1}

{27,1,8}

{27,8,1}

Please check out my answer