Do six small cubic equations have infinitely many integer solutions?

Prove that each of six cubic equations z² + y²z + x³ + … = 0 of size at most 24, the smallest open cases, has infinitely many integer solutions.

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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "number-theory"
	5	+ ],
	6	+ "type": "proof",
	7	+ "also": [
	8	+ "construction"
	9	+ ],
	10	+ "source": {
	11	+ "key": "frontiermath-open",
	12	+ "name": "FrontierMath: Open Problems",
	13	+ "url": "https://epoch.ai/frontiermath/open-problems/small-diophantine",
	14	+ "id": "small-diophantine",
	15	+ "curator": "Epoch AI",
	16	+ "status": "unsolved",
	17	+ "as_of": "2026-10-20",
	18	+ "catalog": "0521c19bbd38"
	19	+ }
	20	+}