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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
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Given an integer $N$, find solutions to $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$
Here is a method to find your triples that is not rapid, perhaps trivial, and highly conjectural. Hopefully it is still of some interest. First, find positive integers $x$, $y$, and $z$ that solve $ …