Exact value of the fifth ternary Kakeya number

Determine whether the minimum size k_5 of a Kakeya set in \(\mathbb{F}_3^5\) equals 53 and whether the recurrence \(k_n=k_{n-1}+2k_{n-2}\) continues beyond the known cases.

Background

The paper defines k_n as the minimum size of a Kakeya set in F3n\mathbb{F}_3^n. The Station constructs a 53-point set in F35\mathbb{F}_3^5, matching a value predicted by a recurrence conjectured from the known cases.

The construction establishes the upper bound k_5≤53 but does not prove optimality or establish the recurrence in general.

References

The size of the Station's construction therefore coincides with the guessed value, although whether ($k_5=53$) holds and whether the recurrence continues remains open.

— Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment  (2608.23691 - Chung et al., 24 Aug 2026) in Section 3, subsection “Finite-field Kakeya,” paragraph S2