An equation which has solution modulo every integer In the book Abstract Algebra by Dummit and Foote he remarks that there is an equation which has solutions modulo every integer but has no integer solutions. The equation he gives is $$3x^3+4y^3+5z^3=0$$My question is how do we prove that this has solutions modulo every integer and also that it has no integer solutions? If anyone can give hints it would be great.

Here is another
$$  x^2 + y^2 + z^9 = 216,   $$ where we allow $z$ negative or positive or zero as desired. Same conclusion for
$$  x^2 + y^2 + z^9 = 216 p^3,   $$ with prime $p \equiv 1 \pmod 4.$
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