Show that the Diophantine equation $x^3+117y^3 = 5$ has no solutions.

I tried using like an odd and even argument for $x$ but it doesn't seem to work because it doesn't matter if $x$ is odd or even. Any help would be great.


1 Answer 1


Hint: The cubes modulo $9$ are $0$, $1$, and $-1$.

  • $\begingroup$ That is really clever! How did you know to use modulo 9? $\endgroup$ Dec 4, 2015 at 20:18
  • $\begingroup$ You could also use modulo $3$ I believe. $\endgroup$ Dec 4, 2015 at 20:39
  • $\begingroup$ Conceivably. But not directly, since modulo $3$ our equation becomes $x^3\equiv 2\pmod{3}$, which does have a solution. $\endgroup$ Dec 4, 2015 at 21:26

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