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arXiv · 2603.29831

On the shortest open cubic equations

Bogdan Grechuk·Ashleigh Ratcliffe
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Abstract

We use cubic reciprocity to prove that the equation $7x^3+2y^3=3z^2+1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations.

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Bogdan Grechuk, Ashleigh Ratcliffe. 2026-03-31. On the shortest open cubic equations. https://arxiv.org/abs/2603.29831

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