How do you show that the equation $x^3+y^3+z^3=1$ has infinitely many solutions in integers? How about $x^3+y^3+z^3=2$?
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You can reduce the first equation to $$x^3 = -y^3, z = 1$$ with obvious infinite solutions. This paper details other families of solutions.
The second equation has solutions $(x,y,z)\equiv (6t^3+1, 1-6t^3, -6t^2)$ which (AFAIK) you find by construction (i.e you have to guess it).
For $x^3+y^3+z^3=1$ it is trivial - an infinite family of solutions is $(1,n,-n)$, and permutations of that.
For $x^3+y^3+z^3=2$ I'm not so sure there are infinitely many solutions. Are you just hypothesizing this, or do you know it to be true?