<s>[INST] Show that $x^3+y^3 = 3\mod 9$ has no solutions Not even sure how to go about this. I tried $x^3 = 3-y^3 \mod 9$, but not sure what that does.
 [/INST] Modulo 9, we have:
$$0^3=0, 1^3=1, 2^3=-1, 3^3=0, 4^3=1, 5^3=-1, 6^3=0, 7^3=1, 8^3=-1$$
No pair of these sums to 3.
In fact, we can use this table to conclude the stronger statement that $$x^3+y^3+z^3=4\pmod{9}$$ has no solutions.
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