I know for any pairwise co-prime integers $x,y,z$ that $$x^3+y^3+z^3\neq 0$$ $$x^3+y^3+3z^3\neq 0$$ Do we also have $$x^3+y^3+2z^3\neq 0?$$ Any Hints?
Does the equation $x^3+y^3+2z^3=0$ have any non-trivial pairwise co-prime integer solutions?
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diophantine-equations
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0Ok! But you should change the title please! – 2017-01-24
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1Trivial answers as mentioned below: $(x, y, z) \in \{(a,-a,0), (a,a,-a)\}$. In these cases clearly $x,y,z$ are not coprime so these don't qualify. – 2017-01-24
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0What would you suggest? – 2017-01-24
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2... have any pairwise co-prime integer solutions? – 2017-01-24
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0http://www.artofproblemsolving.com/community/c3046h1056636_diophantine_equation_3rd_degree – 2017-01-24
1 Answers
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There are no non-trivial solutions. A proof that there are no non-trivial solutions of the closely related equation:
$$x^3+y^3=2z^3$$
is given in Sierpinski's Elementary Theory of Numbers, Chapter 2 page 79, which may be accessed here (which I found via here). Then it just needs to be noted that a solution of either one of these equations would provide a solution of the other by change of sign of $z$.