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.
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Hint: The cubes modulo $9$ are $0$, $1$, and $-1$.
André Nicolas
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That is really clever! How did you know to use modulo 9? – Zubin Mukerjee Dec 04 '15 at 20:18
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You could also use modulo $3$ I believe. – user19405892 Dec 04 '15 at 20:39
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Conceivably. But not directly, since modulo $3$ our equation becomes $x^3\equiv 2\pmod{3}$, which does have a solution. – André Nicolas Dec 04 '15 at 21:26