By Fermat Last theorem, I don't know if that's been discussed.
Find all positive integers $x,y,z$ such $$x^3+y^3=3z^3$$
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4$\begingroup$ This is a genus-$1$ curve over $\mathbf{Q}$ with a rational point $[1:-1:0]$ isomorphic over $\mathbf{Q}$ to the elliptic curve $E: y^2-y=x^3-\frac{1}{3}$ with algebraic rank $0$ (according to MAGMA). $\endgroup$– user19475Commented Dec 5, 2018 at 6:28
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2$\begingroup$ Using Approach0 I found this post on Mathematics about this equation: Three variable, third degree Diophantine equation. $\endgroup$– Martin SleziakCommented Dec 5, 2018 at 6:33
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1$\begingroup$ For a generalisation: Which primes are sums of two cubes? people.mpim-bonn.mpg.de/zagier/files/mpim/95-61/fulltext.pdf $\endgroup$– user19475Commented Dec 5, 2018 at 6:35
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1 Answer
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This is proved in Hardy and Wright, (An introduction to the theory of numbers), Theorem 232. (After proving first the classical $x^3+y^3=z^3$, the notation and methods are ready, to prove this case in a quite similar way.)
answered Dec 5, 2018 at 9:40